People can have as many postal addresses as they wish. Mail is delivered as addressed, not according to the name on the mail, explains U.S. Postal Service publication 508. There is no need to request permission from the USPS to use multiple mailing addresses.
However, there is a difference between receiving mail at multiple residences as opposed to using a single mailing address to establish residency. Residency is a state-mandated requirement for voting and paying income tax that requires a person to establish a primary residence and mailing address. Generally, a person is required to live in a state for a given amount of time before establishing residency. A person can own a residence in a state, such as a vacation home or hunting cabin, while having official residency status in another state.