Uiua, 35 bytes

⬚0↯:°⊚▽≠⍜⊂≡(⧻⊜π.⍜⬚∞⊡¬)⊙(,¤⟜⊚)⟜:⊂.⧻.

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⬚0↯:°⊚▽≠⍜⊂≡(⧻⊜π.⍜⬚∞⊡¬)⊙(,¤⟜⊚)⟜:⊂.⧻.
                                      ⊂.⧻. # push [length length], keeping original
                                   ⟜:      # duplicate around the original
                          ⊙(,¤⟜⊚)         # between the [len len]s:
                               ⟜⊚          #   push the location of the 1s,
                              ¤             #   a 'fixed' version of the array
                             ,              #   and duplicate the locations of the 1s
         ⍜⊂                                # push [len len] to the locations
            ≡(           )                  # for each location:
                   ⍜⬚∞⊡¬                   #   set its cell in the fixed array to 0
              ⧻⊜π.                         #   and count the number of groups of 1s
         ⍜⊂                                # pop the count for [len len]
       ▽≠                                  # keep the points that aren't equal to [len len]'s count
    °⊚                                    # convert them to a binary array
⬚0↯:                                       # pad to size [len len] with 0s

Python3, 341 bytes

E=enumerate
def G(g):
 while g:
  q,s=[(t:=[*g][0])],[t];g-={t}
  for x,y in q:
   for X,Y in[(1,0),(0,1),(0,-1),(-1,0)]:
    if(u:=(x+X,y+Y))in g:s+=[u];q+=[u];g-={u}
  yield{*s}
def f(b):
 B=[[0 for _ in i]for i in b]
 for i in G({(x,y)for x,r in E(b)for y,v in E(r)if v}):
  for j in i:
   if len([*G(i-{j})])!=1:x,y=j;B[x][y]=1
 return B

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Java 8, 503 489 459 455 407 bytes

int R,C,v[][];m->{int r[][]=new int[R=m.length][C=m[0].length],i=R*C,a=i(m),x,y,q;for(;i-->0;m[y][x]=0,r[y][x]=i(m)!=a?1:0,m[y][x]=q)q=m[y=i/C][x=i%C];return r;}int i(int[][]m){int r=0,i=R*C,x,y;for(v=new int[R][C];i-->0;)if(m[y=i/C][x=i%C]>v[y][x]){d(m,x,y);r++;}return r;}void d(int[][]m,int c,int r){v[r][c]=1;for(int k=-3,x,y;k<4;k+=2)if((x=c+k%2-k/2)>=0&x<C&(y=r+k/2)>=0&y<R&&m[y][x]>v[y][x])d(m,x,y);}

-66 bytes thanks to @ceilingcat.

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Explanation:

int R,C,                    // Amount of rows/columns on class-level
    v[][];                  // Visited-matrix on class-level

m->{                        // Method with int-matrix as both parameter and return-type
  int r[][]=new int[R=m.length][C=m[0].length],
                            //  Create the result-matrix, and set `R` and `C`
      i=R*C,                //  Index-integer
      a=i(m),               //  The current amount of islands
      x,y,                  //  Temp integers for the coordinates
      q;                    //  Temp integer for the value
  for(;i-->0                //  Loop `i` over each cell:
      ;                     //    After each iteration:
       m[y][x]=0,           //     Temporarily change the current value to 0
       r[y][x]=i(m)!a?      //     If the current and initial amount of islands are not
                            //     the same anymore:
                1           //      Set the result-value at this coordinate to 1
               :            //     Else:
                0,          //      Set the result-value at this coordinate to 0
       m[y][x]=q)           //     Then change the current value back to what it was
    q=m[y=i/C][x=i%C];      //   Save the current value at this coordinate
  return r;}                //  Return the result-matrix

// Separated method to determine the amount of islands in a matrix
int i(int[][]m){
  int r=0,                  //  Result-count, starting at 0
      i=R*C,                //  Index integer
      x,y;                  //  Temp integers for the coordinates
  for(v=new int[R][C];      //  Reset the visited array
      i-->0;)               //  Loop over the cells:
    if(m[y=i/C][x=i%C]      //   If the current cell is a 1,
       >v[y][x]){           //   and we haven't visited it yet:
      d(m,x,y);             //    Check every direction around this cell
      r++;}                 //    And raise the result-counter by 1
   return r;}               //  Return the result-counter

// Separated method to check each direction around a cell
void d(int[][]m,int c,int r){
  v[r][c]=1;                //  Flag this cell as visited
  for(int k=-3,x,y;k<4;k+=2)//  Loop over the four directions:
    if((x=c+k%2-k/2)>=0&x<C&(y=r+k/2)>=0&y<R
                            //   If the cell in the direction is within bounds,
       &&m[y][x]            //   and it's a path we can walk,
         >v[y][x])          //   and we haven't visited it yet:
      d(m,x,y);}            //    Do a recursive call for this cell

Javascript 122 bytes

Input/output as a multiline string.

m=>m.replace(/./g,(v,p,m,n=[...m],f=p=>n[p]==1&&(n[p]=0,v=f(p-1)+f(p+1)+f(p-w)+f(p+w)-1?1:0,1))=>(f(p),v),w=~m.search`\n`)

For each walkable cell, put a block and try to fill the 4 neighbor cells. If the current cell is not a cut point, then starting from any open neighbor will fill all of them. Else, I will need more than one fill operation to reach all neighbor cells.

Less golfed

m=>{
  w = m.search('\n') + 1; // offset to the next row
  result = [...m].map( // for each cell
     ( v, // current value
       p  // current position
     ) => {
     n = [...m]; // work on a copy of the input
     // recursive fill function from position p
     // returns 1 if managed to fill at least 1 cell
     fill = (p) => {
        if (n[p] == 1)
        {
           n[p] = 0;
           // flag will be > 1 if the fill from the current point found disjointed areas
           // flag will be 0 if no area could be filled (isolated cell)
           var flag = fill(p+1) + fill(p-1) + fill(p+w) + fill(p-w);
           // v is modified repeatedly, during recursion
           // but I need the value at top level, when fill returns to original caller
           v = flag != 1 ? 1 : 0;
           return 1; // at least 1 cell filled
        }
        else
           return 0; // no fill
     }
     fill(p)
     return v // orginal value or modified by fill function
  }) 
}

Test

var F=
m=>m.replace(/./g,(v,p,m,n=[...m],f=p=>n[p]==1&&(n[p]=0,v=f(p-1)+f(p+1)+f(p-w)+f(p+w)-1?1:0,1))=>(f(p),v),w=~m.search`\n`)

var test=`in:
11101001
11011101
00000001
11101111
11110101
00011111
10110001
11111111
out:
01000000
00001001
00000001
00000101
00110000
00010000
00000000
11100000

in:
1111111111111111
1000000000000001
1111111111111101
0000000000000101
1111111111110101
1000000000010101
1011111111010101
1010000001010101
1010111101010101
1010101111010101
1010100000010101
1010111111110101
1010000000000101
1011111111111101
1000000000000001
1111111111111111
out:
0000000000000000
0000000000000000
0000000000000000
0000000000000000
0000000000000000
0000000000000000
0000000000000000
0000000000000000
0000000000000000
0000000000000000
0000000000000000
0000000000000000
0000000000000000
0000000000000000
0000000000000000
0000000000000000

in:
1011010001111010
1111111011101101
1110010101001011
1111001110010010
1111010000101001
0111101001000101
0011100111110010
1001110011111110
0101000011100011
1110110101001110
0010100111000110
1000110111011010
0100101000100101
0001010101100011
1001010000111101
1000111011000010
out:
0000000000111010
1011110001001000
0000000000000011
0000000100010000
0000010000101000
0000001000000100
0000000011000000
1001100000011110
0000000001000010
0110100001000110
0000100101000010
1000100000000000
0100001000000100
0000000100100001
0000010000111000
0000010000000010
`.match(/\d[10\n]+\d/g);
for(i = 0; test[2*i]; ++i)
{
   input = test[2*i]
   check = test[2*i+1]
   result = F(input)
   ok = check == result
   console.log('Test '+ i + ' ' + (ok?'OK':'FAIL'),
   '\n'+input, '\n'+result)
}

Perl 5, -p0 105 101 96 93 90 89 bytes

Uses b instead of 1 in the input.

Make sure the matrix on STDIN is terminated with a newline

#!/usr/bin/perl -p0
s%b%$_="$`z$'";s:|.:/
/>s#(\pL)(.{@{-}}|)(?!\1)(\pL)#$&|a.$2.a#se&&y/{c/z />0:seg&/\B/%eg

Try it online!

Uses 3 levels of substitution!

This 87 byte version is both in input and output format easier to interpret, but is not competing since it uses 3 different characters in the output:

#!/usr/bin/perl -0p
s%b%$_="$`z$'";s:|.:/
/>s#(\w)(.{@{-}}|)(?!\1)(\w)#$&|a.$2.a#se&&y/{c/z />0:seg&/\B/%eg

Try it online!

It's easy to save another byte (the regex s modifier) in both versions by using some different (non-alphanumeric) character as row terminator (instead of newline), but that makes the input quite unreadable again.

How it works

Consider the substitution

s#(\w)(.{columns}|)(?!1)(\w)#c$2c#s

This will find two letters that are different and next to each other horizontally or vertically and replace them by c. In a maze whose paths consist of entirely the letter b nothing will happen since the letters are the same, but as soon as one of the letters is replaced by another one (e.g. z) that letter and a neighbor will be replaced by c and repeated application is a flood-fill of the connected component with c from seed z.

In this case I however don't want a complete flood-fill. I want to fill only one of the arms neighboring z, so after the first step I want the z gone. That already works with the c$2c replacement, but I later want to restart a flood-fill along another arm starting from the same point and I don't know which of the cs was originally z anymore. So instead I use

s#(\w)(.{columns}|)(?!\1)(\w)#$&|a.$2.a#se

b | a is c, b | c is c and z | a is {. So in a maze with paths made up of b and a seed z in the first step b will get replaced by c and z will get replaced by { which is not a letter and does not match \w and so will not cause further fills. The c however will keep a further flood-fill going and one neighbor arm of the seed gets filled. E.g. starting from

  b                      c
  b                      c
bbzbb       becomes    bb{bb
  b                      b
  b                      b

I can then replace all c by some non letter (e.g. -) and replace { by z again to restart the flood-fill:

  -                      -
  -                      -
bbzbb       becomes    cc{bb
  b                      b
  b                      b

and repeat this process until all neighbors of the seed have been converted. If I then once more replace { by z and flood-fill:

  -                      -
  -                      -
--z--       stays      --z--
  -                      -
  -                      -

The z remains behind at the end because there is no neighbor to do a transformation with. That makes clear what happens in the following code fragment:

/\n/ >                                    

Find the first newline. The starting offset is now in @-

s#(\w)(.{@{-}}|)(?!\1)(\w)#$&|a.$2.a#se

The regex discussed above with @{-} as number of columns (since plain @- confuses the perl parser and doesn't properly substitute)

&&

The /\n/ always succeeds and the substitution is true as long as we can still flood-fill. So the part after && is executed if the flood-fill of one arm is done. If not the left side evaluates to an empty string

y/{c/z / > 0

Restart the flood-fill and return 1 if the previous flood-fill did anything. Else return the empty string. This whole piece of code is wrapped inside

s:|.: code :seg

So if this is executed on a starting string $_ with a z at the seed position the piece of code inside will be executed many times mostly returning nothing but a 1 every time a neighbor arm gets flood-filled. Effectively $_ gets destroyed and replaced by as many 1s as there are connected components attached to z. Notice that the loop needs to be executed up to sum of component sizes + number of arms times but that is OK since it will "number of characters including newlines * 2 + 1" times.

The maze gets disconnected if there are no 1's (empty string, an isolated vertex) or if there are more than 1 arms (more than 2 1s). This can be checked using the regex /\B/ (this gives 0 instead of 1 on older perl versions. It's arguable which one is wrong). Unfortunately if it doesn't match this will give an empty string instead of 0. However the s:|.: code :seg was designed to always return an odd number so by doing an & with /\B/ this will give 0 or 1.

All that is left is walking the whole input array and at each walkable position seed with z and count connected arms. That is easily done with:

s%b%$_="$`z$'"; code %eg

The only problem is that in the non-walkable positions the old value is retained. Since we need 0s there that means the original input array must have 0 in the non walkable positions and 0 matches \w in the original substitution and would trigger flood-fills. That's why I use \pL instead (only match letters).

MATL, 26 bytes

n:"GG0@(,w4&1ZIuz]=~]vGZye

The input is a numerical matrix, using ; as row separator.

Try it online! Or verify all test cases.

Explanation

n           % Implicit input: matrix. Push number of elements, N
:           % Range: gives [1 2 ... N]
"           % For each k in [1 2 ... N]
  GG        %   Push input matrix twice
  0@(       %   Write 0 at position k (in column-major order: down, then across).
            %   The stack now contains the original matrix and a modified matrix
            %   with 0 at position k
  ,         %   Do twice
    w       %     Swap
    4       %     Push 4. This specifies 4-element neighbourhood
    &1ZI    %     Label each connected component, using the specified
            %     neighbourhood. This replaces each 1 in the matrix by a
            %     positive integer according to the connected component it
            %     belongs to
    u       %     Unique: gives a vector of deduplicate elements
    z       %     Number of nonzeros. This is the number of connected components
  ]         %   End
  =~        %   Are they different? Gives true of false
]           % End
v           % Concatenate stack into a column vector
GZye        % Reshape (in column-major order) according to size of input matrix.
            % Implicit display

Stax, 40 bytes

Çóê↓â.Φ}╞│*w<(♦◙¼ñ£º█¢,D`ì♥W4·☺╛gÇÜ♠╗4D┬

Run and debug test cases

This program takes input as a space separated string containing rows. The output is in the same format. Here's the unpacked ascii representation.

{2%{_xi48&GxG=-}_?m}{'1'2|e{"12|21".22RjMJguHgu%

The fundamental operation for counting an island works like this.

1. Replace the first '1' with a '2'.
2. Regex replace '12|21' with '22'.
3. Split on spaces.
4. Transpose matrix.
5. Repeat from 2. until a string is repeated.
6. Repeat from 1. until there is no longer a '1' in the string. The number of repetitions is the number of islands.

.

{               start map block over input string, composed of [ 01]
  2%            mod by 2. space and 0 yield 0. 1 yields 1. (a)
  {             start conditional block for the 1s.
    _           original char from string (b)
    xi48&       make copy of input with current character replaced with 0
    G           jump to unbalanced }, then return; counts islands (c)
    xG          counts islands in original input (d)
    =           are (c) and (d) equal? 0 or 1 (e)
    -           b - e; this is 1 iff this character is a bridge
  }             end conditional block
  _?            execute block if (a) is 1, otherwise use original char from string
m               close block and perform map over input
}               goto target - count islands and return
{               start generator block
  '1'2|e        replace the first 1 with a 2
  {             start generator block
    "12|21".22R replace "12" and "21" with "22"
    jMJ         split into rows, transpose, and rejoin with spaces
  gu            generate values until any duplicate is encountered
  H             keep the last value
gu              generate values until any duplicate is encountered
%               count number of iterations it took

Bonus 44 byte program - this version inputs and outputs using the grid format.

Wolfram Language (Mathematica), 118 bytes

(r=ReplacePart)[0#,Cases[#~Position~1,p_/;(c=Max@MorphologicalComponents[#,CornerNeighbors->1<0]&)@r[#,p->0]>c@#]->1]&

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Python 2, 290 bytes

lambda m:[[b([[C and(I,J)!=(i,j)for J,C in e(R)]for I,R in e(m)])!=b(eval(`m`))for j,c in e(r)]for i,r in e(m)]
def F(m,i,j):
	if len(m)>i>=0<=j<len(m[i])>0<m[i][j]:m[i][j]=0;F(m,i,j+1);F(m,i,j-1);F(m,i+1,j);F(m,i-1,j)
b=lambda m:sum(F(m,i,j)or c for i,r in e(m)for j,c in e(r))
e=enumerate

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-11 bytes thanks to Rod
-11 bytes thanks to Lynn