Hey Moira, I think you've touched on a UK/USA difference here. As a Canadian, I never, ever say "have" in question tags after "have" when it's indicating possession. I always use "do" like so:
You have a car, don't you?
And formally,
You have a vehicle, do you not?
When "have" is functioning as an auxiliary, such as in the present perfect, I would use "have" in the question tag:
You've seen that movie, haven't you?