Musing about this thread this morning over coffee and wondering if it's maybe not the "My" that implies possessiveness to people, but the role.
"Mother" would never imply a possessive relationship. It would be absurd to argue with siblings that, no, she's MY mother (despite the emphasis on the "my"). And it would be absurd for other people to claim that she is their mother, unless there is some close motherly relationship there (although most people will say in those cases that she is "like a mother to me").
"Friend" is another similar term, although you do get the "but she's MY friend" people from time to time (had that in college once - had to stifle the WTF hard, since it was my roommate who pulled that).
Husband/wife (or BF/GF, partner, etc.) *can* indicate possessiveness for those who are inclined to do so. And if you believe that the role (wife, for example) dictates that you have authority over her as "husband", then it doesn't matter how you string the words together - when someone talks about their husband/wife, you'll read things through your own experience, at least until challenged.
Dunno. I haven't finished my coffee yet, so the brain's just moseying along this one. FWIW, I don't find MY use of "my" to be possessive, but to indicate a relationship or closeness. "The man I'm partnered with" sounds more distant to me than I'd like.
However, when talking about MY coffee? Yeah. Pry it outta my cold, dead fingers, people.
Mono. Divorced, two kids (DanceGirl, 15; and PokéGirl, 12), two cats, one house, many projects.
My partner. Poly. In relationships with me, Xena, and Noa.
Poly. In a relationship with Chops. Dating others.
Married, Poly. In a relationship with Chops.
External blog with a mono/poly focus: From Baltic to Boardwalk