The ISP I use at home also provides access points in cafés etc. Can they, for example with my MAC address, match my identity when I'm using one of these access points to when I'm at home? And what steps would one take to avoid this?
Identification via MAC address is only possible, if your ISP can "see" that. MAC addresses are only seen on the client that are in the local ethernet collision domain, i.e. if they're on the same switch (simply put). This, of course, includes the router that acts as a gateway.
So, if at home you have a router that your ISP can't access, your MAC can't be seen beyond that router. At a public access point, which acts as a router, it can be seen by the owner of that AP - likely that ISP.
That being said, your ISP is likely to be able to recognize you in different café's, but not at home. In order to avoid this, you'd have to spoof your MAC, which is not easy and requires root access.
That much about MACs, someone who wants to identify you is very likely to use some other kind of identification that can be tracked across routers. Cookies, browser fingerprinting, just to name two. There're more than that and it's hard to counter all of them.
One thing I'd like to add:
Though tracking by ISPs is an issue, it's not the kind of tracking that'd worry me most. ISPs are usually specialized on providing you internet access. That's probably because the scope where an ISP can track you, is always limited. As soon as you're in another country or simply logged in to a friend's Wifi, your ISP has little chance of seeing your traffic at all.
On the other hand, if you have, say, a Google or Facebook account that you use across multiple devices - which most people do and of course they're always encouraged to do so, it's much, much easier to use that information for tracking you - and that's only the tracking that's officially done and most likely legal. There're many more trackers out there that have at least a questionable legal status - at least in some countries - and while you actually can refrain from using Google or Facebook (though avoiding them completely is difficult), it is even harder to avoid trackers that you may not know or think of.
Remaining completely anonymous on the internet is a very hard thing. TOR may be a good choice against just about anything but the NSA, but there're issues, too, and you have to know what you're doing.
Yet one more thing:
Most trackers do not know who you are. Usually they do not care. They're not going to write you a letter, so they don't need your address - letters are too expensive anyway. Same thing for phone calls. The police may look for that if they think they have to, but usually that's about it.
For a tracker it's enough to know something like that: "This person - likely male, probably married with an annual income somewhere around $100.000 has been searching the net for information about pregnancy tests. Therefore he might be interested in buying those as well as baby toys, baby clothing, baby insurance, maybe a bigger car and so on, so if that person loads one of our advertisements, it might be a good idea if it's from one of those categories. Since he lives somewhere around Syracuse, N.Y., we may also add some stores for baby supplies from that area." Try it. Google those things a bit and watch your banners miraculously change. It worked in my case. :)