The notion of session is supposed to follow the user, or, more precisely, the user's browser on a given machine. It is not supposed to be specific to a tab or windows, because tabs and windows (in the usual Web architecture) are a local concept which the user opens and closes at his discretion. By definition, the session also spans several successive pages.
In order to implement the notion of session, many sites push into the browser (as a hidden form field, a cookie, URL rewriting or some other technique) a session token which does not change from page to page (because it has no need to change). This is normal. One can imagine a session-management system where what is stored client-side morphs from page to page, but that would be more complex, with no clear advantage over a simple unchanging session token.