I wonder what other arrangement it could even possibly constitute.

Bailment? That would mean physically locking the bills that you deposit in a safe that you rent, which is possible I guess, but not what we’re talking about here. Trust? This would mean the deposit does not go on the bank’s books, and they cannot use it for their own purposes. This is clearly not the case, at the very least since investment banks and savings banks were merged. Agency? That would mean the bank uses your money to enact transactions on your behalf, again, clearly not the case.

That leaves the only other form of “I give you money and you give it back later”, name debt.