From the PHP docs: "A visitor accessing your web site is assigned a unique id, the so-called session id. This is either stored in a cookie on the user side or is propagated in the URL."
When you log out, that cookie is expired.
Even if you don't log out, destroying the cookie, when you move to a second computer, the cookie doesn't follow you. So PHP has no way of linking the new session with the old session. If you need to link what you're calling a "session" (as opposed to the PHP $_SESSION), then you'll need to store all of that data somewhere yourself. PHP won't do it for you.
You can write out the $_SESSION variables to a database when the user logs out if you have an explicit logout function. If you need a user to be able to just wander away and resume later without having explicitly logged out, then you'll probably need to save all of that information on each page the user accesses.