Yes, if the 2nd page's session_id is the same as the first's, then the session cookie is being properly propagated and received. You might want to try some really simple experiment to see if there is something odd with the configuration or environment.
page1.php:
<?php
ini_set('display_errors', 1);
error_reporting(E_ALL);
session_start();
$_SESSION['test'] = "This is a test.";
?>
<p><a href='page2.php'>Go to next page</a></p>
page2.php:
<?php
ini_set('display_errors', 1);
error_reporting(E_ALL);
session_start();
print_r($_SESSION['test']);
?>
<p><a href='page1.php'>Back to page 1</a></p>