Grammatical possession does not imply property. "My grandmother," "my hometown," "my skeleton," etc.
In Chapter 21 of the Screwtape Letters, CS Lewis discusses the dangers of interchanging of the various flavors of "my". (Well, a fictional demon monologues about the utility of confusing "my" in the mind of humans, but that's the Screwtape Letters for you.)
I would say the skeleton is a property of the body. I wouldn't say I'm my skeleton - I'm just a flesh automaton animated by neurotransmitters...
"flesh automaton" reminds me of this great short story by Terry Bisson I came across in an issue of MIT's Twelve Tomorrows