Mandatory Banking

Here's a question I've been pondering that I'd like to throw out for the economists - my understanding is that although people tend to think of the money in their bank account as 'theirs', in practice they have 'lent' the money to the bank who then lend it to other people but promise to give it back to the lender if they need it (as often noted, a promise they can only keep under certain conditions). So really the money is only 'theirs' if they keep it under their mattress or something.

Now, it seems that in the modern world, for all intents and purposes, having a bank account is pretty much vital if you want to do anything at all - e.g. try getting your employer to pay your salary if you don't have a bank account - so we now live in a society where lending money to banks is (for all intents and purposes) mandatory. I have a feeling this should change our whole concept of what money is and several of the fundamental principles underpinning it. So: How does 'mandatory' banking affect the theory of money?

(originally a comment on this metafilter thread)