The "natural" concept of property is much more narrow than our abstract notions. Animals, and human societies before they develop the abstract property theory, generally define property through possession or use - your house is yours because you live in it, that sort of thing. You can't establish an association simply by saying, "this is mine". This puts a natural limit on how much property can be accumulated.
Once you have abstract property as a concept, it's all about tracking the association in and of itself, separated from any physical manifestations of it (possession and use). Your house is yours not because you live in it, but because there's a title on it in your name, and a record in the registry. This makes it possible for one person to have ten houses, live in one, and rent the rest out to those who don't have any. If they accumulate the rent, they can eventually buy more houses, and it becomes a positive feedback loop.
But there's no particular reason why any given society has to adopt such a system of property, and not all societies did or do today - especially with respect to land and other immovables. This can often be seen in the colonial history, where the colonizers expected to be able to buy and sell things that the indigenous population didn't even consider property, or at least the kind of property that could be bought and sold. More often than not, the colonizing power would then declare that in the absence of a "proper" abstract property claim on e.g. the land, it's there for the taking of anybody who is willing to make one. And that is how we ended up in the present situation, where that system of abstract property rights is mostly enforced throughout the globe - but I don't see how it follows that it's a system that is inevitable for any human society.