If you live in a house you can generate tens of thousands of addresses that will be delivered to you without much problem.

You just begin inventing apt. numbers - having up to a thousand apts in a building is not suspicious. And once you cross a thousand, you can invent new buildings by strapping letter names to your home number, so that gives you a-z * 1000 = 24000 unique addresses you can generate easily. Without much hassle you could extend that to a million.

Requiring a unique mailing address is just as pointless as a unique email.