Ah, another case of Magic vs. Science.
I personally think that seeing as magic is fictitious and so is this game, that it can easily be explained with magic. The major items you get are magical and can be bound to you and the Song of Time was created to take you back as well as magical items in order to complete your quest. Items like rupees and stuff don't go with you so that you don't abuse it for monitary gain. But then... the bank must be magical too, right? That part is the only hole in my theory I think. As for the bank who knows. Maybe it could be explained as they update links rupees on his hand or something like that lol. Then he would be dipping into funds that he hasn't put in there.