While this might seem esoteric at first, a more inclusive question that probably gets closer to the quick of what the OP is trying to learn is to ask "who is the bearer of [BLURGH] ?" which simply means "who holds the title of [BLURGH]?". This is actually a more useful questions because the "rent-or-own?" idiom is deliberately nebulous because so often we say we "own" our car or our house when in fact the entity that really owns it is the financing agency who holds the title until we have paid it off. However, since asking that level of detail would make you seem like a bit of an inquisitor, social mores have us accept that vagueness since the "own" infers that the "owner" is still paying it off, otherwise they probably might proudly say "it's paid off!", but this gets into a whole other area.
Not to derail the OP's question, my point is that when asking someone if they "rent or own?" (which in my mind almost always implies a person's physical residence and fails to address who really owns it) it is unclear whether the OP is just looking for a fancy way to ask "rent or own?" or is the point of the OP's question to clearly determine the status of the person's relationship to the property that they appear to be in possession of? I am inferring that it's the latter that the OP seeks considering that "rent or own?" is so idiomatic it really needs no further explanation or embellishment. Therefore, if it's the hard facts that the OP seeks, "rent or own?" not only fails to address title status, it also doesn't include the possibility that the person is squatting on the property, that the item might be borrowed or stolen, or other imaginative scenarios that might have that person in possession of said [BLURGH].
Sorry if this sounds pedantic, but I thought this 'beared' further discussion. I hope it helps!