If you see a session as "a browser session", then this is surprising behaviour. But this is not the case.
A session is a session as defined by the server. To remember that this is that same user, it saves the session as a cookie. For the point of view of the server, it doesn't really matter if you close your browser, shut down your computer, or drink a cup of coffee: you are still that same, unique, person, so your session should be the same.
As long as your cookies are saved AND are not too old, it's all the same session. You could, from the user side, try to instruct your client to stop this, for instance on a shared account: have the browser remove all cookies on exit or use different profiles (this at least is possible in chrome).
So the expected behaviour is that as long as the cookie is valid, the session is the same. Cookie validity (or actually: removal) CAN be tied to closing your browser, but most of the time it isn't. I am not sure it is even possible to directly detect if a browser was closed, so it's hard if not impossible to force your described behaviour from the server side.
edit: a quick addition from the manual:
Session support in PHP consists of a way to preserve certain data across subsequent accesses. This enables you to build more customized applications and increase the appeal of your web site. All information is in the Session reference section.
It talks about "subsequent accesses", which is quite broad.
var_dump($_SESSION['admin'])and check what is thr in it?session_set_cookie_params(0); and thensession_start();