JavaScript (Node.js), 102 bytes

s=>'aeiou'.replace(r=/[aeiou]/ig,(c,j)=>s.match(c+'|'+(w='AEIOU'[j]))?s.replace(r,d=>d<{}?w:c)+`
`:'')

Try it online!

Perl 5 MList::Util=uniq -n, 74 bytes

@;=uniq lc=~/[aeiou]/ig;s/[aeiou]/$b/g&s/[AEIOU]/\U$b/g&say while$b=shift@

Try it online!

Y is not a vowel in this implementation.

Japt v2.0a0 -R, 24 22 19 bytes

Treats y as a vowel. Change both occurences of \y to \v to treat it as a consonant.

f\y â £r\y_¶u ?X:Xv

Try it

f\y â £r\y_¶u ?X:Xv     :Implicit input of string U
f                       :Matches of
 \y                     :  RegEx /[aeiouy]/gi (\v is /[aeiou]/gi)
    â                   :Deduplicate
      £                 :Map each X
       r\y              :  Replace /[aeiouy]/gi in U
          _             :  Pass each match through the following function
           ¶            :    Is equal to
            u           :      Itself uppercased
              ?X        :   If true return X
                :Xv     :   Else return X lowercased

Java (OpenJDK 8), 184 bytes

y is a vowel

s->s.chars().mapToObj(c->""+(char)(c>96?c:c+32)).filter("aeiouy"::contains).distinct().forEach(c->System.out.println(s.replaceAll("[aeiouy]",c).replaceAll("[AEIOUY]",c.toUpperCase())))

Try it online!

Java 10, 196 188 bytes

s->{var d=new int[99];for(var c:s.toUpperCase().replaceAll("[^AEIOU]","").toCharArray())if(d[c]++<1)System.out.println(s.replaceAll("[AEIOU]",c+"").replaceAll("[aeiou]",(char)(c+32)+""));}

-8 bytes thanks to @joH1.

Without y as vowel to save bytes.

Try it online.

Explanation:

s->{                       // Method with String parameter and no return-type
  var d=new int[99];       //  Integer-array indicating which vowels we've already output
  for(var c:s.toUpperCase()//  Convert the input to uppercase
            .replaceAll("[^AEIOU]","")
                           //  Remove all non-vowels
            .toCharArray())//  Convert it to a character array)
                           //  And loop over those vowel-characters
    if(d[c]++              //   Increase the vowel-count by 1
             <1)           //   And if it was 0 this iteration:
      System.out.println(  //    Print with trailing newline:
        s                  //     The input,
         .replaceAll("[AEIOU]",c+"")
                           //     with every uppercase vowel replace with the current vowel
         .replaceAll("[aeiou]",(char)(c+32)+""));}
                           //     and every lowercase vowel replaced as well

R, 138, 129 bytes

function(x,U=utf8ToInt,a=U(V<-'aeiouAEIOU'))for(i in (which(a%in%U(x))-1)%%5)cat(chartr(V,intToUtf8(rep(a[i+c(1,6)],e=5)),x),'
')

Try it online!

• y is not considered a vowel

Red, 229 bytes

Taking y for a non-vowel

func[s][v: charset"aoeiu"w: charset"AOEIU"p: copy""parse s[any[[copy c[v | w](if not find p c[append p c lowercase c
parse s[any[[copy n to[v | w | end]](prin n)opt[v(prin c)|[w(prin uppercase copy c)]| skip]]]print""])]| skip]]]

Try it online!

Slightly more readable:

f: func [ s ] [
    v: charset "aoeiu"
    w: charset "AOEIU"
    p: copy ""
    parse s[ 
        any [
            [ copy c [ v | w ]
                ( if not find p c [ 
                    append p c
                    lowercase c
                    parse s [
                        any [
                            [ copy n to [ v | w | end ] ]
                                 ( prin n )
                            opt [ v ( prin c )
                                | [ w ( prin uppercase copy c ) ]
                                | skip
                                ]
                            ] 
                        ]
                    print ""
                ] )
            ]
            | skip
        ]
    ]
]

05AB1E (legacy), 19 bytes

(Indirectly) saved one byte thanks to Kevin (printing directly inside the loop rather than joining, only work in the legacy version).

lžMÃÙεžMDu«s5×Du«‡=

Try it online!

Using the Elixir rewrite, 20 bytes

lžMÃÙεžMDu«s5×Du«‡}»

Try it online! (without y) | Try it online! (with y, žM being replaced by žO – same applies for the legacy version)

How it works

lžMÃÙεžMDu«s5×Du«‡}»     Full program. Example: "Hello"
l                        Convert the input to lowercase. "Hello" –> "hello"
 žMÃ                     Keep only lowercase vowels. "hello" –> "eo"
    Ù                    Remove duplicates. "eo" –> "eo"
     ε            }      For each of the characters (example with "e"):
      žMDu«              Yield "aeiouAEIOU"
           s5×           Swap, and repeat the current char 5 times. "e" –> "eeeee"
              Du«        Duplicate, uppercase and merge. "eeeee" –> "eeeeeEEEE"
                 ‡       Transliteration. For each item in B, replace it in A with
                         the corresponding item in C.
                   »     Join on newlines.

Perl 6, 64 bytes

{m:g:i/<[aeiou]>/».lc.unique».&{$^v;say S:g:ii[<[aeiou]>]=$v}}

Try it online!

JavaScript (Node.js), 99 bytes

Treats \$y\$ as a consonant.

s=>(g=F=>Buffer(s).map(c=>2130466>>c&c>64?F(c):c)+`
`)(v=>g[v&=31]||(g[v]=S+=g(c=>c&96|v)),S='')&&S

Try it online!

Commented

s => (                   // s = input string
  g = F =>               // g = helper function taking a callback function F
    Buffer(s)            // turn s into a Buffer
    .map(c =>            // for each ASCII code c in s:
      2130466            //   2130466 is a vowel bitmask: 1000001000001000100010
                         //                               u     o     i   e   a
      >> c               //   the ECMAScript specification enforces that the shiftCount is
                         //   the result of masking out all but the least significant 5 bits
      & c > 64           //   also make sure to ignore non-letter characters
      ?                  //   if a vowel is identified:
        F(c)             //     invoke F with c
      :                  //   else:
        c                //     just yield c
    ) + `\n`             // end of map(); coerce back to a string and append a newline
  )(v =>                 // invoke g with a callback that takes v:
    g[v &= 31] || (      //   unless this vowel has already been encountered:
      g[v] =             //     mark it as encountered
      S +=               //     and append to the output string S
      g(                 //     the result of another call to g:
        c => c & 96 | v  //       where vowels are replaced with v, using the original case
      )                  //     end of inner call to g
    ),                   //
    S = ''               //   start with S = ''
  ) && S                 // end of outer call to g; return S

Python, 129 119 112 bytes

import re
f=lambda s:'\n'.join(r('[AEIOU]',v.upper(),r('[aeiou]',v,s))for v in'aeiou'if v in s.lower());r=re.sub

Try it Online!

Doesn't treat \$y\$ as vowel.

-7 bytes thanks to @Mr.Xcoder

Python 2, 112 bytes

s=input()
V='aeiouAEIOU'
for v in V:
 if v in s.lower():print''.join([c,[v.upper(),v][c>'Z']][c in V]for c in s)

Try it online!

Treats y as a consonant.

bash, 96 bytes

Two equal-length solutions:

v=aeiouAEIOU;for x in `grep -o [$v]<<<$1|sed 's/./\L&&&&&\U&/'|awk !a[\\$0]++`;{ tr $v $x<<<$1;}
v=aeiouAEIOU;for x in `tr -cd $v<<<$1|sed 's/./\L&&&&&\U&\n/g'|awk !a[\\$0]++`;{ tr $v $x<<<$1;}

Try it online!

Takes input as a command line argument and outputs to STDOUT.

Jelly,  23 20 18  17 bytes

-2 Thanks to Erik the Outgolfer

ØcŒHZx5fƇðØc,yð€Y

To treat y as a vowel replace both cs with ys.

Try it online!

How?

ØcŒHZx5fƇðØc,yð€Y - Link: list of characters, S
Øc                - vowels -> "AEIOUaeiou"
  ŒH              - split in half -> ["AEIOU", "aeiou"]
    Z             - transpose -> ["Aa", "Ee", "Ii", "Oo", "Uu"]
     x5           - times 5 -> ["AAAAAaaaaa", "EEEEEeeeee", "IIIIIiiiii", "OOOOOooooo", "UUUUUuuuuu"]
        Ƈ         - filter keep if:
       f          -   filter keep only -> those of X which have required vowels
                  -                       ...i.e. if S = "blah" then ["AAAAAaaaaa"]
         ð    ð€  - dyadic chain for €ach:
          Øc      -   vowels -> "AEIOUaeiou"
            ,     -   pair       e.g. ["AEIOUaeiou","AAAAAaaaaa"]
             y    -   translate  e.g. swap A for A, E for A, ...
                Y - join with newlines

Retina, 45 bytes

~(K`A\EI\OU
L$`\\?(.)
./$1/i&$*\T`Vv`5*$&$L$&

Try it online! Does not count y as a vowel. Explanation:

K`A\EI\OU

Replaces the text with the literal string A\EI\OU.

L$`\\?(.)

Matches each letter optionally preceded by a backslash.

./$1/i&$*\T`Vv`5*$&$L$&

Outputs a line of Retina code for each letter.

~(

Evaluates the generated code (shown below) using the original input. The . causes the code not to output the (final) buffer. The /<vowel>/i& causes the rest of the line to run only if the input contains the given vowel (case-insensitively). The * causes the result of the line to be ignored, so that the next vowel can be tested. The \ causes the result to be printed on its own line before it is ignored. The T`Vv`AAAAAa transliterates uppercase Vowels to AAAAAs and all lowercase vowels to a. \A is an escape that refers to ASCII 07 (BEL), but E, O and o are built-in character classes that needs to be escaped to give their literal values (e is not a character class, but fortunately it is also not an escape.)

./A/i&*\T`Vv`AAAAAa
./E/i&*\T`Vv`\E\E\E\E\E\e
./I/i&*\T`Vv`IIIIIi
./O/i&*\T`Vv`\O\O\O\O\O\o
./U/i&*\T`Vv`UUUUUu

Ruby, 78 bytes

->s{s.downcase.scan(/[aeiou]/).uniq.map{|v|s.tr"AEIOUaeiou",v.upcase*5+v}*?\n}

Try it online!

A quick and naive approach. Y is not considered a vowel.