If the purchase goes well, the vehicle is reliable, spare parts are easily available, and nobody has to resort to legal action? No problem.

On the other hand, if I buy a car in Holland, get it serviced in Austria, and register and drive it in Slovakia - which legal system governs the transaction? Which country's consumer rights protections do I get?

It would be jolly inconvenient if the manufacturer ripped me off, and my only option for redress was a court 1000km away, with the laws written and courts operated in a language I didn't speak.

Or does the EU have some mechanism that avoids this problem?

Do the remote service teams travel across the border? That would probably be more impactful than the physical distance if e.g. there was some minor recall that travelling repair men could take care of while the car is parked at home or work.