WordPress login and php project

WordPress does not use session variables by default. It could be some plugins or themes do, but not core WP. User authentication is managed with cookies. If your code is under the same domain as WP, your code is sent cookie data with the request, but it’s not that simple to verify that the cookie is still valid. You are better off making a custom Ajax or admin-post.php request to WP asking it to verify the logged in status.