Do you mean in the same position, or just the same characters? Your examples suggest the former, but your code the latter. Well, for the former, simple enough, this would be my idea:

  ($word1 ^ $word2) =~ tr/\0//
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Do a bitwise XOR of the words. There where they have the same letters, the resulting character will be a "\0". Next, count these occurrences.

As for the latter, even though it looks simpler as a requirement, as an implementation, it is not. So I'll take a completely different approach.

my %bag;
my $total = 0;
foreach(split '', $word1) {
    $bag{$_}++;
}
foreach(split '', $word2) {
    if($bag{$_}) {
        $bag{$_}--;
        $total++;
    }
}
return $total;
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In English: cut the first word into characters. Put them in a bag. For each character in the second word, pick it out of the bag, at least, if there's at least one of it left in the bag. Count the number of characters taken from the bag.