I think you may have misunderstood. PHP uses PHPSESSID to store the ID of a session. Normally this value is stored in a cookie, but it can also be stored in the url if cookies are disabled.
If you read the value of the url (or the cookie) on one browser, you can use that value in the url in another browser to effectively take over that session. That is because PHP doesn't track any information, but just uses that session id to identify a session.
I guess in your case, cookies are enabled as well, so PHP uses the stored cookie rather than the URL value. You can try to delete the cookie first, disables cookies altogether and use this url, or you can change the value of the cookie.