Different ideas of ownership are getting intertwined here. One is to do with property, but thta's not the only thing we mean when we say "my".

I think it was Chesterton (it normally is) who wrote a passage pointing out that when we say my boots, and my wife, and my country and my God, we aren't exactly using "my" in the same sense. Or at least we shouldn't be. My song is another case, and it fits better closer to the end of that list than the beginning. It's not so much it belongs to you as you belong to it.