"How do I use the map command for this?"

No idea. :-)

"I am pretty sure I need to use the map command"

Why?

In the code below, I simplified the data so that it's easier to see what's happening. I've also jumbled it up to demonstrate the two sorts — you may not need or want either of these (biological data can be huge, so don't do any sorting unless it's absolutely necessary).

#!/usr/bin/env perl

use strict;
use warnings;

my %res;

{
    local $/ = "\n>";

    while (my $record = <DATA>) {
        chomp $record;
        $record = substr $record, 1 if $. == 1;
        $record =~ s/\s*\z//m;
        my ($id, $anno_seq) = split /\|/, $record, 2;
        push @{$res{$anno_seq}}, $id;
    }
}

print '>', join('|', (sort @{$res{$_}}), $_), "\n"
    for sort { $res{$a}[0] cmp $res{$b}[0] } keys %res;

__DATA__
>id7|anno2
GHI
>id1|anno1
ABC
>id6|anno1
ABC
>id4|anno2
GHI
>id2|anno3
DEF
>id3|anno1
ABC
>id5|anno1
ABC
[download]

Output:

>id1|id3|id5|id6|anno1
ABC
>id2|anno3
DEF
>id4|id7|anno2
GHI
[download]