Technically, both are correct. However, if your grammar book or teacher is telling you that "Have you a cat?" is a normal way to ask a question in modern English, you should get rid of them immediately! It is extremely outdated, and sounds wrong to the modern ear. No native speaker has asked this question in this way for a long time, although you may come across this type of question in books published before about 1960 (!) or hear it in plays and films set in previous centuries. What you can say, though, is "Have you got a cat?". This idiomatic form is a common alternative to "Do you have a cat?", and can be used in any informal or neutral situation.

Hi , the difference is that the first one is correct and the second one is incorrect .(At least in American English. Not sure about British or other kinds of English ) "Have you a .... " pattern is not used but we understand what it means .

They're both essentially the same question: you're asking someone else if they have a cat. The difference is that absolutely nobody words it like in the second sentence. It sounds very archaic. So while both are "correct," you will raise a few eyebrows if you say "have you a cat?" instead of "do you have a cat?"