The problem is you are not looking for unique words in each file, but for unique words across all the files. Do I understand your requirement correctly?

#!/usr/bin/perl
use warnings;
use strict;

my %seen;

for my $file (@ARGV) {
    open my $FH, '<', $file or die $!;
    while (<$FH>) {
        chomp;
        s/,\s*$//;
        push @{ $seen{$_} }, $file;
    }
}

for my $string (grep 1 == @{ $seen{$_} }, keys %seen) {
    print "$string unique in $seen{$string}[0]\n";
}
[download]

$ perl 1077045.pl [123].txt
cat unique in 1.txt
dog unique in 3.txt
rat unique in 2.txt
[download]

If the word is considered "unique" even if it occurs several times in one file, you might need to use a HoH instead of a HoA.