They are, in fact, a government agency. Government agencies that are funded entirely through user fees -- which is what postage is -- aren't entirely uncommon. (Of course, given the way that it is funded in part by "loans" from the Treasury that are made without any indication that the Postal Service will ever have a business plan that enables them to be repaid, its kind of questionable to even put the Postal Service in the category of purely user-fee funded government agencies. Its really just a plain old regular tax-and-user-fee-funded government that has been cloaked in some misleading smoke-and-mirrors.)

No, they're an independent government agency. Federal agencies that survive on non-appropriated funds aren't uncommon, for example, The Federal Reserve, Office of the Comptroller of the Currency, U.S. Mint and several DoD NAFIs (e.g. the different exchanges and various MWR programs).