It only occupies a small ephemeral spot in spacetime. Try lying in a single space for hours or days on end in a city and see how far your clever countering gets you. I'm sure there's many a vagrant who would rebut your thesis.
I wouldn't think of my body as private property. Leaving that aside, the answer to your question is that everybody has a body, so we're not talking about the same kind of privilege that car owners have. Nor do bodies occupy the same amount of space. Last but not least, they don't easily kill other people as they cross their way.
But unless you are totally naked, your body (not your private property) is covered by your private property and/or carries it.
It is clear that a car occupies a lot more space than your clothes and carry-on belongings, but as said earlier that is only quantitative and not qualitative.
And cars don't kill people (yet), their drivers may.
