There is a difference between the economics and the game model.
How does it benefit the farmer if you pick their crops?
Economically, they are paying you for your labour. They presumably sell the crop for more than they pay you for picking them.
Economically, yes, I think it is safe to assume that the farmer 'owns' the crops as he would have tilled the land and planted them. If you don't give him the crops you harvest, it is effectively stealing from him.
The game seems to have a simple economic model for farming. You can happily take crops as ingredients or "sell" them to the farmer. I haven't seen any evidence that the farmer will on-sell his crops to retailers or what not. I also haven't done a price comparison of wholesale vs. retail. Perhaps an economic major will create a mod one day.