I was not aware of the multiple value per hash key.

Technically, there's only one: A reference to an array. Just like for AoAs. References are scalars, so they can be stored in arrays and hashes.

It will not always have the same number of elements

Which one is the cat?

I'm not sure how to use your method to output my final data to a file

It already does. The output is written to the $fh_out. I skipped over opening the file handle, but there's nothing special to it:

my $file_out = 'data.out';
open(my $fh_out, '>', $file_out)
   or die("Unable to create output file $file_out: $!\n");
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File handle one will be variable in elements.

For example in one file input it could be

goat1=dog1=bird1=cat1
or
tiger2=goat2=dog2=bird2=cat2

Because of this
   my ($dog, $bird, $cat1) = split /=/;

will be a problem since in this case the code assumes there will always be $dog, $bird, $cat, while there could be more.


The last element however, will always be "cat" while the ones before will change.

File 2 will remain the same even though file 1 can change.
File 2 is
cat1=cat3
cat3=cat4 ...

Variable length ordered list? Array!

use strict;
use warnings;

my $file_in1 = '...';
my $file_in2 = '...';
my $file_out = '...';

open(my $fh_in1, '<', $file_in1)
   or die("Unable to open input file $file_in1: $!\n");

open(my $fh_in2, '<', $file_in2)
   or die("Unable to open input file $file_in2: $!\n");

open(my $fh_out, '>', $file_out)
   or die("Unable to create output file $file_out: $!\n");

my %cat2s_by_cat1;
while (<$fh_in2>) {
   chomp;
   my ($cat1, $cat2) = split /=/;
   push @{ $cat2s_by_cat1{$cat1} }, $cat2;
}

while (<$fh_in1>) {
   chomp;
   my @fields = split /=/;
   my $cat1 = $fields[-1];

   exists( $cat2s_by_cat1{$cat1} )
      or die("Cat1 $cat1 not found in file2\n");

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