https://bugs.winehq.org/show_bug.cgi?id=39891
--- Comment #13 from Omar gamingpakker+wine@gmail.com --- I have not tested with a wired 360 controller as I do not have one. I'm asking around right now if I can find someone that has one.
What you describe with the wired 360 controller is the proper behaviour. That is how the game has always functioned for me on Windows with the wireless 360 controller as well. Iirc non-Xbox controllers show the same behaviour as your generic USB Joystick on Windows as well. I'll double check with a generic joystick as well, I should still have one laying around somewhere.
This does sound like there may be a discrepancy between how the wired vs wireless xbox controllers are eventually handled within WINE?