I know that prendi means ‘take’ but I am wondering what it would mean in this context.
Tu, vieni qui, prendi casa?
It also seems it can be used in other ways?
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This sentence is a bit weird, but in this context it should mean "You should come (move in) here and rent (or buy!) an apartment (or house). It has nothing to do with visiting anyone. "Prendere" can have several meanings depending on the context, among them to buy, rent or order. "Casa" can stand for "house", but also "home", so, the sentence has indeed somewhat romantic and poetic tone, but the explanation of the context you provided is way off.