Your chop'ing the element in @file2 waaaay too many times. Had you use the recommended chomp, all you have have lost is CPU cycles. With chop, you're loosing data.

use strict;
use warnings;

my $file1 = shift;
my $file2 = shift;

open(my $fh1, '<', $file1) or die "Cant open $file1: $!\n";
open(my $fh2, '<', $file2) or die "Cant open $file2: $!\n";

chomp( my @file1 = <$fh1> );
chomp( my @file2 = <$fh2> );

my %counts;
for my $base (@file1) {
   my $re = qr/^\Q$base\E(\.|\z)/;
   for my $node (@file2) {
        ++$counts{$base} if $node =~ /$re/;
   }
}

for my $base (keys(%counts)) {
   print("$base: $counts{$base}\n");
}
[download]

or even

...

my %counts;
for my $base (@file1) {
   my $re = qr/^\Q$base\E(\.|\z)/;
   $counts{$base} += grep /$re/, @file2;
}

...
[download]