If I am asking someone what he got for his birthday, would it be
"What did you get?"
"What have you got?"
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To ask someone what was given to them on their birthday
What did you get?
What did you receive?
are the two common questions
What have you got?
would be incorrect since the receiving is in the past.
Also be aware that
What have you got? = What do you have?
which is a different meaning then you might have intended.