When you have 1 dollar in your account, the bank still has the dollar! It's a liability owed to you.

Every financial asset is somebody else's liability. Every financial asset is similar to a promissory note. If you sign one, it is your liability, but it would be an asset for the note's holder. This is also true for your bank account. Your account statement is a promissory note that the bank owes you 1 dollar.