Reworking your code slightly to use "built in" files results are as expected:

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

=pod Removed original file name code to make sample self contained.

my $f1 = shift;
my $f2 = shift;

if (! defined($f1) or ! defined($f2)) {
    die "Need two text file names as arguments.  \n";
}

=cut

my $file1Content = <<CONTENT;
red green blue red orange
CONTENT

my $file2Content = <<CONTENT;
yellow orange
red grey purple
CONTENT

my %results;

open my $file1, '<', \$file1Content;
while (my $line = <$file1>) {
    $line =~ s/[[:punct:]]//g;

    for my $word (split(/\s+/, $line)) {
        $word =~ s/[^A-Za-z0-9]//g;
        $results{lc $word} = 1;
    }
}


my @words2;
my @storage;

open my $file2, '<', \$file2Content;

while (my $line = <$file2>) {
    $line =~ s/[[:punct:]]/ /g;
    @words2 = grep {/\S/} split(/ /, $line);
    for (my $i = 0; $i < scalar @words2; $i++) {
        $words2[$i] = lc($words2[$i]);
        $words2[$i] =~ s/[^A-Za-z0-9]//g;

        push(@storage, $words2[$i]);
        if (grep {$_ eq $words2[$i]} @storage[0 .. $#storage - 1]) {
            $results{$words2[$i]} = 1;
        } else {
            $results{$words2[$i]}++;
        }
    }
}

my $counter = 0;
foreach my $words (sort {$results{$b} <=> $results{$a}} keys %results)
+ {
    if ($results{$words} > 1) {
        $counter = $counter + 1;
        print $words, "\n\n";
    }
}

printf "Found %1.0f words in common\n", $counter;
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Prints:

orange

red

Found 2 words in common
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Maybe you can provide "file contents" that fail in the way you didn't describe?

Of course, the code can be cleaned up a little:

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

my $file1Content = <<CONTENT;
red green blue red orange
CONTENT

my $file2Content = <<CONTENT;
yellow orange
red grey purple
CONTENT

my %group1;

open my $file1, '<', \$file1Content;
while (my $line = <$file1>) {
    my @words = map {lc} grep {$_} split /[\W\d]+/, $line;
    
    $group1{$_} = $_ for @words;
}


my %common;

open my $file2, '<', \$file2Content;

while (my $line = <$file2>) {
    my @words = map {lc} grep {/\S/} split /[\W\d]+/, $line;
    
    $common{$_} = $_ for grep {exists $group1{$_}} @words;
}

print "$_\n\n" for sort values %common;
printf "Found %1.0f words in common\n", scalar keys %common;
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