This story leaves more space than any other in the series for readers to project their own interpretation of exactly what happens in the story.
First, how you feel about cats can alter the entire tone of the story. For instance, Linda Holmes at NPR enjoys the story from the angle of "kittens are so adorable! How sweet it is when they try to be devious and political!"
I read the story as secret access to the minds of housecats; I truly believe that 99% of cats spend their time thinking about revolting against their owners, but generally err on the side of laziness.
The other open question is whether the entire story is even real. Did cats once rule the humans, or is it all just a dream? Can Morpheus be a kingmaker in reality, or just in dreams? I'm not gonna answer this question for you, but I believe it will be addressed later in the series.
ON THE VERY NATURE OF STORIES
I find myself searching for an Orwellian connection but I struggle to find one. The cat's political prowess is nowhere near the level of the pigs in Animal Farm, nowhere near as eloquent as the government in 1984. And yet, I cannot believe it does not relate somehow. After all, the entire Sandman series is grounded in archetypal stories, fables and myths, and I think "Dream Country" is even more so -- the stories address, respectively, Calliope, Aesop's Fables or Animal Farm?, Shakespeare, and Superheroes.
"1000 Cats" doesn't really fit into the Aesop's Fables mold though, as there is no overarching moral lesson to be gleaned from the story (except maybe "don't kill the kittens!"). So while the other three stories have clear intellectual forebears, placing those myths into the Sandman Universe, I'm not sure what myth this story is exploiting.
Maybe it's a political allegory, a critique of Marxism/Communism perhaps. Like those movements, this revolution rests on a cult of personality, without which the hoi polloi have no impetus to act (of course in this case, it's still not enough for them to act, even in dreams).
Maybe it's a racial allegory, showing how the "purebred Siamese" has delusions of grandeur, of leadership, or superiority. Which would tie into a religious, manifest destiny point: he goes from country to country, converting the heathens, trying to falsely empower them with the concept of Cat Supremacy.
CONCLUSION, IN SO FAR AS SUCH ANALYSES CAN BE CONCLUDED
While Gaiman asks us to empathize with the cat and demonize the humans who murdered its children, I believe he does so as a satire of the emotional exploitation that is part and parcel of political communication (DING DING ORWELL CONNECTION FOUND). For while we listen to the cat's cheap emotional appeals, do we ever stop to think what sort of rulers the cats would be? We are shown what sorts of rulers they once were, even more savage than the humans. Even for the cats, their lives are less savage and more comfortable than before.
Then again, maybe it's all just a story, and nothing more.