This is a game's bug. CP2077 always rumbles the first controller (in the enumeration order) but accepts input from all the devices. It behaves the same way on Windows, I've just checked.
Thanks for investigating @ivyl. Since this is game-specific feedback, and we're using one issue report per unofficially supported game title, I've gone ahead and transferred this issue report to https://github.com/ValveSoftware/Proton/issues/4450#issuecomment-905600761.
I've played Cyberpunk 2077 a few days ago and had both my gamepads attached:
I was playing with the wireless gamepad from the couch when I noticed it doesn't rumble at all BUT the wired gamepad on my desk did rumble instead.
I did disconnect the wired gamepad but the wireless gamepad still would not rumble. It required a restart of the game to make it rumble properly.
Expected results: Either the gamepad I'm playing with should rumble, or none should rumble, because gamepads not held by any hand tend to rumble down the desk.