Interesting question. I had never thought about this, but yes it sounds extremely strange to say "I got off the car". I can't really say why there is a difference, but I would always say "I got out of the car", whereas you would usually say "get off the bus" and "get off the plane". Maybe there is an assumption that a car is something you are driving, while with a bus and a plane you are generally a passenger?

"get on" goes with "get off" the same way "get in" goes with "get out" -Get on the bus / Get off the bus -Get in the car / Get out of the car.