Here in America people make cars a part of their identity and one of the questions you’ll often hear (maybe after a few conversations) is “Do you drive?”.
I’ve never heard the question “do you drive,” only “what do you drive?” The assumption that every individual owns and drives a car is pretty omnipresent.
Maybe it's a local thing? New York, perhaps? It's definitely not widespread. It'd be a really weird question in most of the country (of course you drive)
There are about 3-5 metro areas in the US where a substantial portion of the population doesn't drive day-to-day by choice (NYC, Chicago, Philadelphia, maybe DC and Boston). I have heard "Do you drive?" asked in those places. Everywhere else, it's "what do you drive?" and "what kind of car are you thinking about paying a bank to buy for you?".
That is something you will hear towny people in the UK ask now. Especially when asking young people who are absolutely putting off learning to drive a thing they will not remotely be able to afford to run.