I'm going to assume that you're asking about Mongol language both spoken and written.
When sedentary civilizations exist long enough to establish a system to write their language, it's because settled societies naturally develop institutions that require writing. As a matter of fact... the general consensus is that writing was first developed by the Sumerians in order to establish accounting to facilitate commerce. Aside from accounting, sedentary civilizations also develop systematic education, scientific exploration, structured commerce, standardized currency, codification of law, etc.
Aside from their complex social dynamics, the life of the average steppe nomad at that time was relatively simple. When you're deemed to be of age, you learn the skills necessary to survive on the steppe. Your wealth was measured based on material possessions, your social success was based on how well-connected you were, and your military success was based on personal magnitude - your ability to rally other people to your side and challenge those who oppose you. There was no need for systematic education, scientific exploration, structured commerce, standardized currency or codification of law. The life of nomadic tribes was so vastly different from settled society that some of the institutions mentioned above would be absolutely foreign to the average Mongol of that time; so much so that their spoken language might have not even had all the words necessary to gracefully describe the inner workings of such institution. For the nomads living on the Mongolian steppe, it was simply survival of the fittest.
Prior to unification under Genghis Khan, the Mongols were illiterate. They did not begin to adopt writing until after Genghis Khan captured an Uyghur scholar named Tata-Tonga who modified the Uyghur script to fit the Mongolian tongue. Most of the nomadic tribes that resided within the boundaries of what we call Mongolia today were also illiterate because their way of life did not require their language to be written. However outside of the Mongol homeland, the major sedentary civilizations (that later fell to the Mongols) already had their own writing systems. In chronological order of conquest: Uyghurs, Tanguts, Jurchens (Jin Dynasty), Khwarazmians, Kieven Rus, Abbasids, and Chinese (Song Dynasty).
There was simply no need to reinvent the wheel. Whatever the Mongols demanded could simply be demanded in the Mongol language and translated to the vassal language. It was simple: We are your overlords and you are our subjects based on right of conquest. We make laws, you follow them, send your troops to aid us when we call, pay your taxes, and don't rebel.
The reason why Latin became the dominant language in the Roman Empire is because the Roman Empire was a sedentary civilization that developed all the institutions mentioned above. Imposing your written language on both literate and illiterate people is not an enormous undertaking when your written language has been established and refined over centuries. The same logic applies to the Arabic Caliphates.
So ultimately, why did the Mongols not impose the use of their own language as a prestige language? It simply was not necessary and could be argued as counterproductive. Also, the Mongol Empire only lasted a little over 150 years. Even if they wanted to make it happen, there was simply too much territory and not enough time.