package Frobcounting;
sub new {
  my ($class, $a) = @_;
  my $h = {};
  $h->{$_}++ for @$a;
  bless $h, $class;
}
sub new_d {
  my ($class, $a, $b) = @_;
  my $h = {};
  my $h = { a => Frobcounting->new($a),
            b => Frobcounting->new($b) };
  bless $h, $class;
}
sub frobcount { $_[0]->{$_[1]} }
sub frobcount_d {
  my ($fc_d, $e) = @_;
  my $ea = $fc_d->{a}->frobcount($e);
  my $eb = $fc_d->{b}->frobcount($e);
  $ea < $eb ? $ea : $eb
}
sub smaller_d {
  my ($fc_d) = @_;
  my @ka = keys %{$fc_d->{a}};
  my @kb = keys %{$fc_d->{b}};
  @ka < @kb ? @ka : @kb
}

package main;

my $fc = Frobcounting->new_d(\@string1, \@string2);
for (sort $fc->smaller_d) {
  print "$_: ${\$fc->frobcount_d($_)}\n"
}
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All that by your problem. As for comparing two arrays, and getting such things as 'the number of elements that exist in both arrays', have you looked at List::Compare ?