Right now, in my pocket, I have coins (of silver, gold, and copper) minted by 6 different outfits. And if I had more coins in my pocket that number of mints represented could go much higher. In my wallet I also have what you would call a "debit card" issued by my bank (backed by my deposited metals), as well as a couple of paper bills (receipts for precious metal) issued by another bank.
Banks are privately owned and are actual secure depositories of their customers' money. They make loans, but only loan out a percentage of the money they possess, and only using the money of customers who have agreed, in exchange for earning interest, to have their money loaned out. (This is the opposite of what was permitted by "fractional reserve banking" in the Era of The State.)
The banks set their own policies for identification. There is absolute privacy for their customers. No reporting deposits to anyone- no questions about where the money came from or what a withdrawal is to be used for. No one, ever, is allowed to take a customer's deposits without the consent of the customer- not even for payment of restitution. No one would continue to do business with any bank that allowed that to happen even once "out of necessity". Not even the person who was getting the money this time. After all, if it happened once, it might happen again, and it might be done to you next time rather than for you.
The paper money that is issued by banks is backed by the metal, in physical form, held by the issuing bank. Most have built-in security features that make counterfeiting difficult and not really worth the effort. (Considering the variety of bills in circulation you would probably remember who gave you the bogus bill anyway.) You can take that receipt to the bank and trade it for the actual metal at any time. Most people prefer to use coins for amounts between 1/10 ounce and 5 ounces of metal, but larger and smaller (of gold and platinum, in particular) weights are frequently paid in bills. This is strictly a personal preference and is by no means universal. Some old-timers still don't trust paper money no matter who backs it- having been burned in the fall of government.
Some people also use money from other parts of the world, and as long as your trading partner is willing to accept it that is no problem. Almost no one will accept money that is not silver, copper, or gold, though. The metal content is what matters, not who minted it. You can use anything as money that anyone agrees to accept in trade. All trade is ultimately barter (it was even during the government era; people just bartered with fiat money back then). You can use silver, bullets, sex, bananas, or labor in trade. No one has any say except you and the person you are trading with. That is the only civilized way to do business.