A simple trick is to do a dummy replacement, ie, replace the matching text with itself, while doing an evaluation on every single match with the 'e' modifier.

#!/usr/bin/perl -w
use strict;

my $text = do { local $/; <DATA> };

$text =~ s/my.*?\((\w+)\)/print "$1\n"; $1/ge;

__DATA__
my dog is (red1)
my cat is (blue1)
my dog is (red2)
my cat is (blue2)
my dog is (red3)
my cat is (blue3)
my dog is (red4)
my cat is (blue4)
my dog is (red5)
my cat is (blue5)
my dog is (red6)
my cat is (blue6)
[download]

And the output is

red1
blue1
red2
blue2
red3
blue3
red4
blue4
red5
blue5
red6
blue6
[download]