With certain exceptions, like the mailbox example, most situations of "rent foo" require the lessor to take a risk, and he wants *some* assurance that he'll get his rented thing back. I'm not going to rent a car to someone without ID, for cash, because I have no guarantee that he isn't going to drive away with it and never come back.
Privacy costs. It is possible to create a company which offers insurance against damage and loss, paid for by the user, assigned to the owner. Such a policy could be presented to a car rental agency in lieu of your name. Your transaction with the rental agency would then be anonymous, even if your transaction with the insurer were not. Such an arrangement might even be preferable to a rental agency, since it means they don't have to go after individuals with shallow pockets in the event of damage or loss. I can even imagine such a company which offers standard policies for any number of different objects, written and digitally signed over the phone. Want to rent and apartment? Get your damage and last month's rent insured. There is already such a thing as "completion insurance" for construction and the like, purchased by the builder as a condition of contract. If the transaction costs of this and similar types of insurance were lowered, anonymity in the real world would increase. Eric