yes, the posessive in english indicates many things, and one possibility is ownership. It can also indicate a link or relationship that is not ownership -- as a child I never thought I owned "my father."

Just bugs me when people look at one meaning of a word or grammatical construction and then assume that's the only meaning.

I agree that we are not our emotions, and I think that's a useful idea to conveny. But I think OOP's take on English grammar is gravely misinformed. Imagine if she had a similar take about a language she didn't know well; she'd be rightly criticized by native speakers.