This kind of session tracking is very likely to be done using a combination of a cookie with a unique id identifying your current session, and the website pairing that id with the last IP address you used to connect to their server. That way, if the IP changes, but you still have the cookie, you're identified and logged in, and if the cookie is absent but you have the same IP address as the one save on the server, then they set your cookie to the id paired with that IP.
Really, it's that second possibility that is tricky to get right. If the cookie is missing, and you only have your IP address to show for identification, it's quite unsafe to log someone in just based of that. So servers probably store additional info about you, LSO seem like a good choice, geo IP too, but User Agent, not so much because they don't really say anything about you, every body using the same version of the same browser as you has the same.
As an aside, it has been mentioned above that it could work with MAC adresses. I strongly disagree! Your MAC address never reaches your bank's server, as they are only used to identify sides of an Ethernet connection, and to connect to your bank you make a bunch of Ethernet connections: from your computer to your home router, or your ISP, then from there to the first internet router you go through, then to the second, etc... and each time a new connection is made, each machine on each side provide their very own MAC addresses. So your MAC address can only be known to the machines directly connected to you through a switch or hub, because anything else that routes your packets will replace your MAC with their own. Only the IP address stays the same all the way.
If MAC addresses did go all the way, it would be a privacy nightmare, as all MAC addresses are unique to a single device, hence to a single person.
This is a slightly simplified explanation because it's not the point of the question, but it seemed useful to clear what looked like a misunderstanding.
