Regardless of your concepts of property someone has to be the arbiter of disputes, which leads back to the need for a formal system for dealing with property.

You can of course always widen the definition of a word to capture anything. Let's for example say all cars are in a public pool. If you sit in the driver's seat you can use the car until you leave the seat. You could now say that the car is your property while you are sitting in the driver's seat or you could say that all people own all the cars. Neither matches the common usages of property well. But you are of course right, it is a resource allocation problem and it will only work well if there are commonly accepted and enforced rules, but there is no need for this system to resemble our current system and understanding of property.