People are free to have as many mailing addresses as they want. The U.S. Postal Service publication 508 says that mail is delivered based on the address, not the name on the mail. You don’t need to ask the USPS for permission to use more than one mailing address.
But there is a difference between using a single mailing address to prove residency and getting mail at more than one place. In order to vote and pay income tax, the state requires people to have a primary home and mailing address. This is called residency. Before becoming a resident of a state, most people have to live there for a certain amount of time. A person can own a place to live in one state, like a vacation home or hunting cabin, even if they live in another state legally.