1953 saw the release of the classic teenage rebellion movie, The Wild One. To modern viewers, its portrayal of "rebellion" might seem a bit quaint. But it helped make Marlon Brando a star, and one line of dialog continues to be repeated as a sort of litany against authority.
Let's suppose Brando had replied with "What do you have?"
Meh.. It just doesn't have the same ring to it.
What if, in the previous sentence, I had written "It just hasn't got the same ring to it"?
It wouldn't be grammatically wrong, but the meaning would be slightly altered.
Both have and got, in a context like this, indicate possession. But have posits the possession itself as a neutral fact. It simply asks the reader/listener to consider the relationship: I have an apple.
Why do I have the apple? Was it because I took some action? Am I glad that I have it? We don't know.
Whereas if I say I've got an apple, the tone changes. That's because got is the participle of to get, and to get is more active than to have.
Getting is about taking possession of something. Having something is just about... well, having it.
If Brando had said "What do you have?" then it would have been like asking: "Do you happen to be associated with any aspects of modern life with which a rebellious young man such as myself might choose to quarrel?"
There's nothing wrong with that, but the alternative is more powerful. "What do you got?" more or less equates to: "You speak as if you have generously invested your time into accumulating information about some ways of life in which I may have the opportunity to rebel. What are they?" This way of talking implies that the world is a place in which rebellion has a great deal of value in itself, and it's a better reflection of Brando's character.
Brando also could have said "What have you got?" This would have been grammatically more correct, but weaker. The use of "have got" (present perfect) would suggest a level of literacy and care in word choice that would have been wrong for the character.