You can change the hardware (MAC) address of your computer’s network adapter, but your ISP knows exactly which IP address is assigned to your location and the source and destination of any traffic you send.
If you’re getting the content from a web site that has the content on it, and the copyright holder or law enforcement can get hold of the transfer logs, they know who accessed what.
For true peer to peer (bittorrent) it’s a little harder. There’s no centralized database to seize, and there’s no exhaustive log of all the transfers made. Some copyright holders have been accused of setting up peer-to-peer seeds of their materials in order to catch file sharers (and if they’re doing it right, they’re not going to use an IP address that’s registered to them directly, so any blocking tools you might find won’t work because you won’t be able to tell the difference between them and anyone else with a generic cable modem or DSL IP address). The copyright holders can join the P2P network, search for their content, and make note of where they find it.
Otherwise, the ISP or the law enforcement agency has to monitor locations in the network, and they’ll only catch anyone whose traffic passes through that point—like watching cars go through an intersection. It’s a pain in the whatever, but it can and is done. Whenever you click “I agree” on an acceptable use policy for an internet service provider, you’re probably granting them permission to log this information and turn it over should they be compelled by law to release it. You have no expectation of privacy.
Copyright holders like to focus on schools because a lot of the sharing is done by students, and the schools have IT departments that can be convinced to help a little more easily than telecoms can be.
You can use encryption and anonymizers to obfuscate your traffic if you want.