To be clear: a company can arbitrarily break something you paid for by terminating your account, or just taking the servers online, that is completely legal. That company can then sue you if you connect to someone else's servers or bring up your own, because that is illegal copyright infringement or contractual violation (and possibly CFAA violation because idiocy).
https://arstechnica.com/gaming/2010/08/the-88-million-server...
And given that, a lawyer could argue that paying to play on (or even donating to) a private server is contributory copyright infringement. I don't know if that has been tested in court.
https://en.wikipedia.org/wiki/Contributory_copyright_infring...
But oftentimes you can get away with it; few game companies go after small private servers.
https://codes.findlaw.com/ca/vehicle-code/veh-sect-21955.htm...