Singleton in PHP not really a Singleton?

Let's say that I have the following singleton class:

==========
<?php
class Singleton
{

      private static $instance;

      private function __construct(){}
      
      public static function getInstance()
      {
            if(!isset(self::$instance))
            {
                  $object= __CLASS__;
                  self::$instance=new $object;
            }
            return self::$instance;
      }
      
      public function displayMessage()
      {
            echo 'I am a Singleton';
      }
}
?>
==========

Now I have two scripts, test.php, and test2.php

They each look like the following:

test.php
==========
<?php

require_once "singleton.php";

$singleton=Singleton::getInstance();
$singleton->displayMessage();// display message

$singleton2=Singleton::getInstance();
$singleton2->displayMessage();// display message

print "$singleton<br />";
print "$singleton2<br />";


      
?>
==========

test2.php
==========

require_once "singleton.php";

$e = new Exception();

$singleton=Singleton::getInstance();
$singleton->displayMessage();// display message

$singleton2=Singleton::getInstance();
$singleton2->displayMessage();// display message

print "$singleton<br />";
print "$singleton2<br />";
==========

Note that test2.php has created an exception object at the top.  So the output of test and test2 is the following:

I am a Singleton
I am a Singleton
Object id #1
Object id #1

I am a Singleton
I am a Singleton
Object id #2
Object id #2

So the Singleton only acts as a 'Singleton' within the current script.  On requests to other pages, it does NOT return the same object id.  On test, the object id is #1, but on test2, the object id is 2.  So this is not really a singleton like in Java?  I don't see how it could be since there is no container.  I thought maybe PHP did something behind the scenes.  And if you put some intialization code in the private constructor, it will run on every getInstance on separate scripts.  So if you have code that connects to the database, it is no different then connecting at the top of every script.  Am I missing something??