Owning a car is definitely not a right. It is not mandatory that you own a car. But you are right on the second half. Hence driver's licenses.
And I honestly see no correlation to the ban cars/ban guns thing. Not even close to the same thing. While cars are used as weapons, it happens a lot less often than a shooting. A lot less, I imagine. I'm sure if using cars as weapons was that "popular," there would be more stories on it on the news.