Beyond the 1893 instance of “Why is a mouse when it spins?” that the OP found in Robert Overton, Ten Minutes, the earliest examples that a Google Books search yields are from 1897. First, from a letter dated January 28, 1897, to The Boston Medical and Surgical Journal (February 18, 1897):
After reading this remarkable paper, one involuntarily recalls a conundrum propounded most seriously by one lunatic to another: “Why is a mouse when it spins?" To which the satisfactory answer was: “The higher the fewer.”
Less than three months later, from the “Roundabouts” section of The American Stationer (May 6, 1897):
Those intellects which at one time were kept in tension by asking: “Who struck Billy Patterson?” and “Why is a mouse when it spins?” and “When is a door not a door?” are now employed on the vital query: “Who was Mother Goose?”
(I have to point out, though, that, unlike the other questions listed in this excerpt, the question “When is a door not a door?” has an obvious answer: “When it’s a jar.”)
And in 1899, from the “Casual Comment” section of Literature: An International Gazette of Criticism (July 14, 1899):
An editorial writer in the Hartford [Connecticut] Courant writes “Why has no one noticed the points of resemblance between Bret Harte and Rudyard Kipling?” This is another one of those mysterious conundrums like “Why is a mouse when it spins?” We do not, of course, know just what our Hartford contemporary has settled upon for the answer, but we cannot refrain from saying that the question reminds us very much of a problem recently put by a learned judge to a distinguished counselor at law, when the two gentlemen found themselves engaged as end-men in an amateur minstrel performance in New York. ...
“Brother Bones,” said the judge, twisting his tambourine airily and with a smile that would have filled the lamented Backus with envious admiration, “why is So-and-so”—naming the middle-man—“the best lecturer in the United States?”
“Why is So-and-so the best lecturer in the United States?” returned the distinguished counsel, repeating the question slowly. “Why, because he ain’t,” he added with a serene confidence that was beautiful to look upon.
Google Books finds seven unique examples of the riddle over the next decade (1900–1910), all in the same nonserious vein.
For example, from John Worne, “Bertie and the Colonel,” in The Sketch (July 29, 1903):
At about twenty past five, Lord Bobby began asking riddles , such as, “If a fish weighs ten pounds and half its own weight, what does it weigh?” or “Why is a mouse when it spins?” ; but they bore it with grim fortitude. Nobody fled.
And from “Parable of the King and the Four Tailors” in The American Tailor and Cutter (1904):
”Ah, Your Royal Joblots! I have a few wheezes for you. Why does the chicken cross the street? Why is a mouse when it spins? How old was Ann? Why is Carrie Nation like a tailor? Aha! That sticks you — eh! — Because she made the saloonkeepers close. But, if you want the real clothes, call on Jack the Joker.”
And finally, from Edward S. Field, “The Purple Stockings,” in McClure’s Magazine (April 1910):
An answering laugh to the grave question, “Why is a mouse when it spins?” is supposed by many to denote a lively sense of humor.
Meanwhile “The higher, the fewer” first shows up in a Google Books search in a match from Susan Gay, Woman and a Future Life (1876) in the context of observed tendencies in number of offspring people of different social classes have:
Fortunately, nature will look after the census of progressive humanity. The higher, the fewer; the lower, the more numerous; as illustrated in the clod-cleaving Hodge, and the very many little Hodges. Look at your own world again, and see what it teaches you. The more elementary the stage in the scale of life, the larger provision for its continued existence.
But the next match is from 1902, almost a decade after Overton’s publication of the riddle (and its answer) in 1893. From Clara Green, “The Emancipation of Theodorus” in Scribner’s Magazine (March 1902):
They [Mrs. Theodorus and “some other devotees of whist”] were a small band who admitted into their circle no one not enlightened on the modern game.
“I suppose they take as their motto : The higher the fewer,” laughed Brewster.
So it appears that “Why is a mouse when it spins?” arose in the late 1800s as a nonsense riddle with an unrelated answer (albeit one that may have had currency as a slogan at and before the turn of the twentieth century)—and that this riddle caught on in much the same inexplicable way that catchphrases like “Has you mother sold her mangle?” and “There he goes with his eye out!” did more than half a century earlier, as reported by Charles Mackay in Extraordinary Popular Delusions and the Madness of Crowds (1841).
Nigel Rees, A Word in Your Shell-like (2004) gives much too late a first occurrence date for the riddle (“remembered from the 1920s/30s”), but provides some useful additional details:
A variation of the riddle (which doesn’t help either) is: Q. ‘Why is a mouse when it’s spinning its web?’ A. ‘Because the more the fewer the quicker.’ Other phrases from nonsensical riddles: Q. ‘How is a man when he’s out?’ A. ‘The sooner he does, the much.’ Q. ‘What’s the difference between a chicken/duck?’ A. ‘One of its legs is both the same.’ Q. ‘Which would you rather, or go fishing?’ A. “One rode a horse and the other rhododendron.’
At least there’s a genuine tradition of these things.