In my mind, of course the server doesn't know if you took a cookie you received from the previous instance of your browser and placed it into the next instance

In my mind, of course the server doesn't know if you took a cookie you received from the previous instance of your browser and placed it into the next instance of your browser, because the act of closing your browser doesn't call every server saying "don't consider this cookie valid ever again," the act of closing your browser just deletes cookies that have a lifetime of 0 (but does not delete cookies with a nonzero lifetime, IIRC).