You might get better performance with stringwise XOR (see perldoc perlop, search for Bitwise String Operators):

sub characters_in_common {
    my ($p, $q) = @_;
    my $r = $p ^ $q;
    my $count = $r =~ y/\0//;
    return $count;
}
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BTW, I don't think your function calculates the Hamming distance...