OK, here we go:
//1) Have you got a car// is semantically identical to //2) Do you have a car//. And the use of one or the other does not necessarily indicate BrE or AmE usage.
The tag answers are: 1) Yes, I have and 2) Yes, I do.
So the point to remember here is that English has two ways of expressing ownership or possession. And they are 100% semantically the same.
It should be noted, however, that in fast speech (when people speak quickly) or informal speech, one often hears in AmE: Got time? Got sugar? Got a car? That is a truncated form of /have you got/: [Have you] got milk? In colloquial speech, there is sometimes a shift of helping verb, so you get this situation:
John: Got a car?
Mary: Yeah, I do.
This shift in helping verb from what should be HAVE in response to "Have you got a/an x" is also heard in BrE.
There is no such thing as BrE HAVE and AmE DO. Americans use /have you got/ and the British use /do you have/. AmE speakers use the truncated form of /[have] you got/ a lot but also use the full-out Have you got x? There is simply no justification to call one form (have you got) British English and the other (do you have) American English.
Question with have: Do you have some sugar? Yes, I have some.
Question with have got: Have you got some sugar? Yes, I have got some. or Yes, I've got some.
Important point: Neither is British or American. Both are semantically the same and both are used in both varieties of English.
Final point: Sometimes people switch (as pointed out above) between the simple have form and the have got form.
Have you got any sugar? Answer: Yes, I do have some.
Do you have any sugar? Answer: Yes, I've got some.
Summary: have and have got are two semantically equivalent ways to express having something (owning or having in one's possession). Each one has an interrogative form. Sometimes, speakers will switch horses in mid-stream and answer the "Do you have some x?" with "Yes, I've got some x?" or "Have you got some x" with "Yes, I have some x."
This has zero to do with British or American usage....