I'm assuming the controller in question isn't actually broken, right? I don't play on PS4, but I don't suppose there's any sort of controller calibration built into the game like the PC version?

I recall that with previous consoles (long ago) I'd encounter a similar problem if the joystick was in a non-resting position when the controller (these were wired, so also the console) was powered on. There's no way that your controller regularly has the joystick pointing up when you turn it on, is there? Seems like it would be rather difficult given the way you turn a PS4 controller on, but that's the extent of my diagnostic knowledge of this problem.