It's not clear what you want to achieve here... Did I understand it right that you want to build a matrix of distances? Then you can just use two nested foreach loops:

use strict;
use warnings;
use Data::Dumper;
use Text::Levenshtein qw(distance);

my $file=$ARGV[0];

my @all_ids=();
open my $in, "<", $file
    or die "Failed to open $file: $!";
while(<$in>)
{
    chomp $_;   
    push @all_ids, $_;
}
close $in;

my @matrix;
foreach my $first( @all_ids ) {
    my @row;
    foreach my $second( @all_ids ) {
        push @row, distance( $first, $second );
    };
    push @matrix, \@row;
};

print Dumper(\@matrix);
[download]

This leaves room for optimization, of course.

P.S. use strict; use warnings; . It's 2017 already!