The USPS used to be a clear-cut government agency, but has been quasi-privatized. Congress cut its taxpayer funding, and directed it to run itself like a business, not like a public service. Accordingly, you are incorrect about who are the "people paying their salaries": the USPS has not received any taxpayer money since its funding was cut to zero in the early 1980s. It now funds itself like a business, by charging fees and, like any other business, exploiting intellectual property where the opportunity arises. In that sense it's more like a regulated utility, like Comcast or PG&E, having been privatized in all but name.

Ultimately whether it's a public service that should act in the public interest, or an organization that should be run as a business, is a policy decision that's up to Congress to decide. And Congress decided on the 2nd. Given that, exploiting the zipcode database as proprietary IP seems in line with the letter and the spirit of their (current) mandate.