It's public and private. It still has to make money to survive because it doesn't get much in the way of tax money.
https://www.uspsoig.gov/focus-areas/did-you-know/postal-serv...
https://www.uspsoig.gov/focus-areas/did-you-know/postal-serv...
For example, just about the only reason Susan Collins is still a congressional representative is because she reliably keeps defense spending rolling into the Portsmouth naval shipyard which in turn is fed by a large network of suppliers.