my %cat2s_by_cat1;

while (<$fh2>) {
   chomp;
   my ($cat1, $cat2) = split /=/;
   push @{ $cat2s_by_cat1{$cat1} }, $cat2;
}

while (<$fh1>) {
   chomp;
   my ($dog, $bird, $cat1) = split /=/;

   exists( $cat2s_by_cat1{$cat1} )
      or die;

   for my $cat2 ( @{ $cat2s_by_cat1{$cat1} } ) {
      print $fh_out (join('=', $dog, $bird, $cat1, $cat2), "\n");
   }
}