The author (Sam Brasch) acknowledges that the Rodent Cheesemakers site is a spoof, but then he seriously addresses the question of how to make cheese from rat milk. He notes that you would need a lot of rats: "You’d need an army of 674 rats to produce the 31 kilograms of milk one dairy cow puts out each day."
But if you had that many rats, they might produce a surprisingly good product:
My thought here: depending on what you fed the rats, their milk might also be quite tasty!
But the International Business Times has also weighed in on this issue and throws cold water on the promise of rat-milk cheese by pointing out, "We don't have milking machines small enough to make rat dairies a viable option."
But they're wrong about this. There are milking machines for rats. Back in 1946, Prof. B.L. Herrington of Cornell University designed a "midget milker" — the world's smallest milking machine mounted on a board 18x6 inches. He designed it primarily to milk guinea pigs, but also used it on rats, rabbits, and hamsters. A Science News Letter article noted that "milking guinea pigs is a two-man operation, with one person holding the animal, and it takes about 10 minutes."
I haven't been able to find any pictures of Herrington's midget milker in action, but there is a diagram of it in a 1951 article in the Journal of Nutrition ("Milking Techniques and the Composition of Guinea Pig Milk").
So there wouldn't be a technological problem with milking rats. It could be done. The problem would be the labor involved. It would take too long to milk enough rats to produce a decent amount of cheese. So it would never be done except as a one-off thing.