I bet the size difference plays a role, and a car turbo that spins to 200k rpm will definetly be smaller than a turbopump for rocket engine.
Yes it does, it matters a lot. With smaller dimensions, tip speeds remain manageable. As soon as parts of the thing get supersonic, everything gets much more complicated.
I'm going to assume the rocket engine turbo has a hell of a lot more mass which makes it a far more difficult problem.