Are there Varieties of Ownership?

Our senior thesis writers this semester have been working on some interesting topics.  Want to raise some questions this week that have come to mind listening to them develop their arguments.  One is looking at the notion of ownership.  From the Enlightenment, rights-based approach to property, there seems to be a single thing which is ownership, but is this true?  Does how you came to own the thing alter what it means to own it?  If four people have the same thing, but one person made it for herself, one person bought it for himself, one person received it as a gift, and one person inherited it from a relative who passed away, the object may have different meanings for them, but is the sense of ownership the same in all of them?

If I buy a Picasso, it is my painting, I own it, but in a sense it is not my painting.  Is there a partial sense of ownership here?  Can something with cultural meaning be fully owned or is it partly owned by the culture?  Does this apply to something like a sports team as well as say an important landmark?