Lexicals don't work (right) in regex code blocks. If you switch to using globals, you'll get the results you are after:

my @strings = qw( aaabbbb ab abb aabb aaabb aabbb );

for my $string ( @strings ) {
    our $a_counter = 0;
    our $b_counter = 0;
    print "In $string there were $a_counter 'a's and $b_counter 'b's. 
+\n"
    if $string =~ /(a(?{$a_counter ++;}))+(b(?{$b_counter ++;}))+/;
}
__END__
C:\test>junk
In aaabbbb there were 3 'a's and 4 'b's.
In ab there were 1 'a's and 1 'b's.
In abb there were 1 'a's and 2 'b's.
In aabb there were 2 'a's and 2 'b's.
In aaabb there were 3 'a's and 2 'b's.
In aabbb there were 2 'a's and 3 'b's.
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There is a warning about this in one of the regex pods.