Well, first off, you should probably always use strict and warnings. use strict would have found your error for you. Well, 1 of the errors.

1. In your final for() loop, you are using an array called @sequence which you never initialize anywhere else. I presume you meant to use @gene there instead.
2. Your $ids have regex meta chars in them (the |), so you need to tell Perl to treat them as normal characters.

If you change your final part to:

for (my $i=0; $i<@genes; $i++)  {
        foreach my $id (@uniq)  {
                if ($genes[$i] =~ /^\Q$id/)       {
                        print "$id\n";
                }
        }
}
[download]

it seems to work.