During WWII the US government built many manufacturing plants to make ammunition, tanks, aircraft, etc. Although the government owned the land and built the buildings, contractors were hired to run them and actually build the weapons and munitions. These facilities are still called GOCO or Government Owned, Contractor Operated.
At the end of WWII, and over the years since, some of these plants have been sold to private companies, and others closed (I believe the old St Louis ammunition plant falls into this category). Some remain open and very active like the Lockheed aircraft plant in Georgia, north of Atlanta where Lockheed has manufactured over the years the C-130s, including the current production of C-130Js, the C-141s, C-5s, and more recently the F-22 fighters. Many of the current buildings at the Lockheed plant have been funded by Lockheed and are not government owned!
The ammunition plant at Lake City is also a GOCO, operated by ATK. ATK pays for all the material it uses to manufacture ammunition at Lake City and sells the completed ammunition to the Federal Government under contracts with the government. It must compete with other qualified companies to win these contracts.
As bdgreen says, ammunition produced by ATK at Lake City that is rejected by the government, and therefore not delivered is still owned by ATK. I suspect they internally sell these “scrap” assets to Federal at Anoka. It is also possible that the ammunition AKMS mentions was actually loaded at Anoka from scrap/surplus cases (and perhaps other components) from Lake City. It is even possible that the cases could have been made for Federal by Lake City, but in that case, there would typically have been some sort of consideration ($s credit or some other consideration) given to the government for any government equipment and/or facilities used. The extra marks on the headstamp may indicate these cases were made specifically for Federal.
Bd green is correct, left over/rejected ammunition from military/government contracts is frequently sold commercially. After WWII, Winchester had millions of rounds of 9mmP ammunition left over from the British Contracts. They sold this ammunition commercially well into the 1950s. Over the past few years I have bought Federal 9mmP left over/rejected from a contract with Taiwan and Speer ammunition left over/rejected from a contract with the French Police.