If you really want to look for any individual common characters between all possible pairs of 2000 sequences then you just gotta do the work, and there is a lot of work to do!

It may help to tell us why you want to do that. There may be a better solution to your problem than the brute force search implied so far.

You may find this code interesting however:

use strict;
use warnings;

my @sequences = qw(ACGCATTCA ACTGGATAC TCAGCCATC);
my %matches;

for my $outer (0 .. $#sequences - 1) {
    for my $inner ($outer + 1 .. $#sequences) {
        my $mask = $sequences[$outer] ^ $sequences[$inner];
        
        next if index ($mask, "\0") == -1; # No matching characters
        
        $mask =~ tr/\0/\xff/c;
        $mask |= $sequences[$outer];
        $mask =~ tr/\xff/./;
        
        push @{$matches{$mask}}, [$outer + 1, $inner + 1];
    }
}

for my $match (sort keys %matches) {
    print "$match pattern between ",
        join (', ', map {"$_->[0] and $_->[1]"} @{$matches{$match}}),
        "\n";
}
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Prints:

.C....... pattern between 1 and 3
.C.G....C pattern between 2 and 3
AC....T.. pattern between 1 and 2
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