Your problem has nothing to do with chomp, but rather the syntax of the match. Refer to the section "Regexp Quote-Like Operators" in perlop.

use strict;
use warnings;
my $secret = \"AA";
close STDIN;
open STDIN, '<', $secret or die $!;




print "Enter a string that will match the secret matching pattern \n";
chomp (my $selection  = <STDIN>);

if ($selection  =~  m/(.)\1/) {
      print "The match was perfect \n"; 
    } else {
      print "Match not found \n";
    }
[download]

OUTPUT:

Enter a string that will match the secret matching pattern
The match was perfect
[download]

Note that you first example only appears to "work". It gets the right answer for the wrong reason.