Try it this way:

use strict;
use warnings;

print "Name:\n";
my $name = <STDIN>;
chomp $name;
open( FH, "names.txt" ) or die $!;
while( <FH> ) {
    chomp;
    if( m/(\d+)\s*\|\s*\Q$name\E/ ) {
        print "$1\n";
    }
}
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