As usual, TIMTOWTDI. However, keeping your initial code structure for better or worse:

use strict;
use warnings;

use Test::More tests => 1;

my $want = <<EOT;
>id1|id2|id3|Q51487|P-474-4|86-98,113-126,297-310,322-335
CSLIPDYQRPEAPVAAAYPQGQAYGQNTGAAAVPAADIGWREFFRDPQLQQLIGVALE
EOT

my %res;

while (<DATA>) {
    if (/^>(.*?)\|(.*)/s) {
        my $id   = $1;
        my $rest = $2;
        if (exists $res{$rest}) {
            $res{$rest}{keys} .= "|$id";
        } else {
            my $seq = <DATA>;
            $res{$rest} = {
                keys => ">$id",
                seq => $seq,
            }
        }
    }
}

my ($k) = keys %res; # You would loop over all keys is your real prog
my $have = "$res{$k}{keys}|$k$res{$k}{seq}";
is $have, $want;

__DATA__
>id1|Q51487|P-474-4|86-98,113-126,297-310,322-335
CSLIPDYQRPEAPVAAAYPQGQAYGQNTGAAAVPAADIGWREFFRDPQLQQLIGVALE
>id2|Q51487|P-474-4|86-98,113-126,297-310,322-335
CSLIPDYQRPEAPVAAAYPQGQAYGQNTGAAAVPAADIGWREFFRDPQLQQLIGVALE
>id3|Q51487|P-474-4|86-98,113-126,297-310,322-335
CSLIPDYQRPEAPVAAAYPQGQAYGQNTGAAAVPAADIGWREFFRDPQLQQLIGVALE
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