How should I know that it was my car? It really could have been anyone's car here. "I have some problem with the car" should be accepted in my opinion.
Several people report that but I can't find any example of such use anywhere.
The translation above is fine, but "...a problem with the car" would be very commonly used, at least in the US East Coast/Mid-Atlantic region. In most cases, the meaning would be understood to be "with my car," though it could easily be used in the sense of, say, "I have a problem with the car that I rented for this trip, and I'm not happy!"
If there's nothing in the Czech sentence that would make "I have some problem with the car" completely wrong, I will add it.