As you know it, 'government cheese' is an inherently funny-sounding phrase we all got a hold of in the 1990's, and then it became increasingly played-out until the only people that would touch it were the most desperate of hack comics.
But how many people got through that entire time without ever actually knowing what government cheese is?
This is government cheese. Its distribution began in 1981, the act of which would soon come under the auspices of the Emergency Food Assistance Program. (PDF file.) The cheese served a dual purpose: in addition to providing food assistance for those on welfare, it also helped solve the government's problem of what to actually do with surplus food supply from the Commodity Credit Corporation- other staples such as powdered milk, butter and flour also come government-issue. Eventually, you want someone eating that food. Better to make a welfare program out of it than to throw it away. (In more recent years, it's also been distributed as part of disaster-relief programs.)
Here is the official USDA guideline on making it. (Another PDF file.) It is made in two- and five-pound blocks, intended for distribution once a month.
What kind of cheese is it? This discussion thread about it on the website Chowhound pegs it as something of a cross between American cheese and Velveeta. That's about what the USDA is going for; according to the guidelines, "Its flavor shall be pleasing and characteristic of process cheese made from mild to medium cured American cheese, and shall be free from undesirable flavors and odors."
Stories from those who've eaten it tend to differ: some like it with nachos or macaroni, others can't wait to get away from the stuff. One person on Chowhound assigned a plastic-like favor to it.
While we're on the subject of government food, there are also two varieties of government peanut butter: one for issue under these same premises, and a second one, called "Standard Reference Material No. 2387", designed to be absolutely scientifically perfect. It costs $220 for a six-ounce jar, but you can't actually buy it because it's used as quality control for the stuff you can buy.
It's probably best if you don't read too much into the implications.