Whether an object is considered to be economically fungible makes no difference to the issue of who owns the object. But maybe that’s beside your point.
On another note, I see no reason why different banks’ bank notes must all be considered equally fungible with one another. If free banking does arise in a free-market, stateless society, I wonder if there might be different exchange rates between the bank notes of different banks and gold (for example). These exchange rates would reflect people’s attitudes about the stability and creditworthiness of the different banks.