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I bought a steam game in POL. It works okay on the first machine.

then I wanted to move the installation to another machine, without
redoing the entire slow Steam install process, so I just copied the wineprefix.

The game started, but wasnt working properly (sound was missing).
Furthermore the game now wants a cd key.

Are Steam games tied to a MAC address or some such nonsense?
Do I miss some settings if I just copy an entire prefix between machines?

Steam itself remembers Mac Addresses and things like that. But that should be your issue. Steam knows when you are logging your account on a different machine, but when that happens all it does is send an unlock code to your email which you then enter into Steam and then it adds the current machine into a safe list for you. It's for account security.

If the game is requesting a CD key I would presume that when you copied over that information you didn't copy over some kind of windows registry file, that's where I would imagine Steam would store the CD key anyways.

If your sound was missing and the new computer has a different sound card, drivers etc it wouldn't surprise me that sound wouldn't work either, but that's probably not a game issue, it's probably an issue with Wine or POL. Try going into the Config area and look for some kind of audio test. Check to see if it plays.

I wish I knew more, but I'm new to POL as well. Good luck.