Just move the my %counts and the print of %counts into into the first foreach:

use strict;
use warnings;

my @site1 = qw(
   AATKKM
   AKTKKM
);

my @site2 = qw(
   GGGGGG
   HHHHHH
);

my $site1;
my $site2;

foreach $site1 (@site1) {
   my @site1_parts = split(//, $site1);

   my %counts;

   foreach $site2 (@site2) {
      my @site2_parts = split(//, $site2);

      my $i = @site1_parts;
      $counts{$site1_parts[$i].$site2_parts[$i]}++ while ($i--);
   }

   print($_, ': ', $counts{$_}, $/) foreach (sort keys %counts);
   print($/);
}

__END__
output
======
AG: 2
AH: 2
KG: 2
KH: 2
MG: 1
MH: 1
TG: 1
TH: 1

AG: 1
AH: 1
KG: 3
KH: 3
MG: 1
MH: 1
TG: 1
TH: 1
[download]

I got rid of MapCar since it wasn't needed. (Yeah, I know I'm the one who introduced MapCar in the first place.) I also switched to data that better illustrates the results.

If you don't want to discard the previous counts before the loops is over, you can use an AoH:

use strict;
use warnings;

my @site1 = qw(
   AATKKM
   AKTKKM
);

my @site2 = qw(
   GGGGGG
   HHHHHH
);

my $site1;
my $site2;

my @count_collection;

foreach $site1 (@site1) {
   my @site1_parts = split(//, $site1);

   my %counts;

   foreach $site2 (@site2) {
      my @site2_parts = split(//, $site2);

      my $i = @site1_parts;
      $counts{$site1_parts[$i].$site2_parts[$i]}++ while ($i--);
   }

   push(@count_collection, \%counts);
}

foreach (@count_collection) {
   my $counts = $_;
   print($_, ': ', $counts->{$_}, $/) foreach (sort keys %$counts);
   print($/);
}

__END__
output
======
AG: 2
AH: 2
KG: 2
KH: 2
MG: 1
MH: 1
TG: 1
TH: 1

AG: 1
AH: 1
KG: 3
KH: 3
MG: 1
MH: 1
TG: 1
TH: 1
[download]