You used the proper namespace here:

$xml->findnodes('/x:ItemLookupResponse/x:Items/x:Item')
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But you forgot here:

$item->findvalue('ASIN')
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Change the latter to

$xml->findvalue('x:ASIN', $item)
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Other issues:

• Your variable names are awful! You call your XML $string while $xml is an XPathContext object for which $xpc is recommended, and you call your document $parser where $doc or $dom make more sense.

• You call load_xml as an object method, but it's a static method like new.

• XML::LibXML::XPathContext isn't loaded by use XML::LibXML; in older versions of XML::LibXML.

Fixed:

#!/usr/bin/perl

use strict;
use warnings qw( all );
use feature qw( say );

use XML::LibXML               qw( );
use XML::LibXML::XPathContext qw( );

my $xml = <<'__EOS__';
<?xml version="1.0"?>
<ItemLookupResponse xmlns="http://webservices.amazon.com/AWSECommerceS
+ervice/2013-08-01">
  <Items>
    <Item>
      <ASIN>B01KI4JSQY</ASIN>
    </Item>
  </Items>
</ItemLookupResponse>
__EOS__

my $doc = XML::LibXML->load_xml(string => $xml, { no_blanks => 1 });

my $xpc = XML::LibXML::XPathContext->new();
$xpc->registerNs('x', 'http://webservices.amazon.com/AWSECommerceServi
+ce/2013-08-01');

for my $item ($xpc->findnodes('/x:ItemLookupResponse/x:Items/x:Item', 
+$xml)) {
    say $item->firstChild->nodeName;
    say $item->firstChild->toString;
    say $xpc->findvalue('x:ASIN', $item);
}
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