I think you'll get a bit further if you use exists to test for uniqueness.

#!/usr/local/bin/perl

use strict;
use warnings;

my @lines = <DATA>;
my (@uniq, @purge, %seen);

for my $line (@lines) {
    
    my ($id) = $line =~ m/ID No: (\d+)/;
    
    if (not exists $seen{$id}){
      push @uniq, $line;
      $seen{$id} = undef;
    }
    else { 
      push @purge, $line;
    }
    
}

print qq{uniq\n};
print @uniq;
print qq{purge\n};
print @purge;

__DATA__
lat="37.4192" lng="-122.0574" United States ID No: 1123631397
lat="37.4192" lng="-122.0574" United States ID No: 1123631397
lat="37.4192" lng="-122.0574" United States ID No: 1123631398
lat="37.4192" lng="-122.0574" United States ID No: 1123631399
lat="37.4192" lng="-122.0574" United States ID No: 1123631400
lat="37.4192" lng="-122.0574" United States ID No: 1123631400
[download]

output

uniq
lat="37.4192" lng="-122.0574" United States ID No: 1123631397
lat="37.4192" lng="-122.0574" United States ID No: 1123631398
lat="37.4192" lng="-122.0574" United States ID No: 1123631399
lat="37.4192" lng="-122.0574" United States ID No: 1123631400
purge
lat="37.4192" lng="-122.0574" United States ID No: 1123631397
lat="37.4192" lng="-122.0574" United States ID No: 1123631400
[download]