1) To answer the "why not" question, partly it is because Marines took pretty much the best tactical ideas from the Mongols, and incorporated them. So a lot of Mongol influence can be seen by proxy.
And partly, it's because the Western audience does not necessarily know that much about Mongols, so using the model explicitly would not help in world-building (and, as per above, all the best stuff of theirs was stolen by later military organizations that ARE used as models explicitly), so not much is lost idea-wise.
2) While I can't at the moment recall any SciFi/Fantasy books using the Mongol Empire as a political model, I know there's at least one book that explicitly uses it as a military model - at least, in terms of tactics.
In Eric Flint's debut novel, "Mother of Demons", the humans - directed by a historian - employ Mongol tactics and probably strategy as well (the book didn't cover enough strategic stuff to be 100% certain).
The night she made her decision to throw their strength to the aid of the Kiktu, Indira had spoken to the little army of human warriors. She had told them the story of the battle of Liegnitz, in a place called Poland.
There, Subedei's Mongols had met the forces of European chivalry under the command of Duke Henry of Silesia. Those forces included knights from all the major militant orders as well as Henry's own troops—Knights Templar, Teutonic Knights, Knights Hospitaler.
She had described the European knights. Heavily armored, dangerous at close quarters. And—very slow; easily confused by any tactics beyond a simple, direct charge.
She had described Subedei's Mongols. Lightly armored; extremely fast; extremely disciplined; shrewd and cunning; well coordinated in battle.
Then she described the battle itself. And told them how the Mongols had cut to pieces the flower of chivalry.
Joseph and his lieutenants had taken over from there. The new tactics which they had been developing recently, with Nukurren's advice, fit perfectly into the plan which they developed for the coming battle.
In general, Eric Flint seems to have a high regard for Mongols. Witness the other quote from the same book, spoken by a hero who pretty much conveys the author's voice - especially see the bolded sections:
Of course, the secret had been known earlier. Much earlier, and by more than one general or people. But never, Indira knew, had the secret been taken closer to heart than by a people whose technology was barely Neolithic. A people who had created the greatest empire in the history of the human race.
As she watched Joseph Adekunle walk away, Indira did not see a tall boy whose ancestors had lived in the rain forest of West Africa. She saw the much shorter and lighter-skinned ghost of a different man, from a different continent.
The maneuvers of that man's armies had been measured in degrees of latitude and longitude, despite the fact that their only vehicles were horses and camels. His soldiers were reputed by his defeated foes to have been an innumerable "horde"; yet, in actual fact, he had been outnumbered in every battle he won. And he won almost all his battles. He had developed principles of discipline combined with lower-level initiative which, to his bewildered and hapless victims, had seemed like magic on the battlefield. He, and his fellow generals, had incorporated the systematic use of artillery into warfare, more than half a millenium before Napoleon. He had, centuries before the invention of electronic communication, discovered the centrality of what a word-besotted later culture would call CCCI— "communication, control, command and intelligence."
His armies, which continued his traditions after his death, shattered every realm which opposed them. China, the most powerful civilization of the epoch, had fallen to them. The cumbersome armies of Europe, moving like snails beneath iron shells, had been slaughtered like lambs. The vastnesses of the Russian forest and people defeated every invader which came against them, throughout history. Except once. Except when they were conquered by armies trained and led by the greatest general the human race ever produced.
Subedei Bat'atur. Born into the Reindeer people, an extremely primitive and obscure tribe related to another obscure and only slightly less primitive tribe, called the Mongols.
Subedei Bat'atur. Commander of the tumens, the 10,000 strong divisions of Genghis Khan's armies.
By his lights, and those of his people, Subedei Bat'atur had been neither cruel nor sadistic. The Mongols simply approached warfare as a practical task, to be carried out as efficiently as possible. The nomads—derided as superstitious savages by the civilized peoples who surrounded them—had, in fact, studied warfare with the clear and unblinkered eyes of a child. They experimented with the tactics and methods of their enemies, and adopted those which they found useful. For all the breathtaking scope of their vision—which was nothing less than the conquest of the entire known world—they were neither haughty nor arrogant. Quite unlike the vastly more cultured Chinese mandarins and the (much less vastly) cultured knights of Europe, who thought there was nothing to be learned from others.
The Mongols taught them otherwise. Or, at least, taught them to fear what they could not understand or learn.
In truth, Indira had always had a certain genuine admiration for the Mongols. The commonly accepted verdict of later history, she thought, was quite unfair. The Europeans who, at the time, had been able to do nothing more than pray for deliverance (which they received, simply because the Mongols, having already conquered half of Europe, decided the other half wasn't worth it) had taken their scholarly revenge centuries later. The Mongols had become synonomous with pure and simple brutality.
How many people knew, Indira wondered, that the Mongols instituted and enforced a policy of religious toleration which was unheard of in the Middle Ages? (She even smiled, then, in that moment of heartbreak, remembering the time that the Great Khan Mongke invited representatives from all the great religions to come to the imperial capital at Qarakorum. They had come—representatives from Islam, Catholicism, Orthodox Christianity, Buddhism, Taoism—and had debated theology before the Great Khan and his court. Gritting their teeth, because the holy men were accustomed to other methods of settling accounts with heretics and nonbelievers. But the debate had not degenerated into violence. Not with the Mongol tumens prepared to enforce the law.)
How many people knew that the Mongols fostered the greatest explosion of trade and commerce that had ever taken place prior to that time between China and the western lands of Islam and Christendom? That they built, in China, twenty-seven observatories—and then invited the world's greatest astronomers to come from Persia to help the Chinese learn to use them? That the Mongols built hospitals and even a medical academy, institutions which would have mystified the Europeans of the time?
Almost no one, thought Indira, beyond a few professional historians. Whatever the glories of their later rule, the methods which the Mongols used to create that rule were all that remained in the common memory of the human race.
An admirable people, in many ways. But they had approached warfare with the clear eyes of a young and unfettered people. They had examined war, and grasped its secret.
Speed above all else. Mobility above all else.
Their victims had numbered in the millions.
P.S. The book, like many other ones published by Baen, is available for free in Baen Free library website, so the quote above is fully legal.