I don't know anything about Mongolia, but I know a bit about China. I am not a historian, so please correct me when I am wrong.
First of all, the idea of a nation-state is European. The Peace of Westphalia in 1648 made it very prominent in (western) Europe, though you can find traces of it before Westphalia. Please read Stephen D. Krasner and Robert O. Keohane if you want to know more about Sovereignty.
In China, before they were introduced to the concept of sovereignty and the nation-state, a concept called Tianxia (All-under-Heaven) was prominent. It was essentially the idea that the world was one beneath the Heaven and that the emperor was the link between Heaven and the world. Thus, he was the highest on Earth and all should obey him. The world was divided on the people closest to the emperor, people further from the emperor, vassal states and barbarians. Now, Tianxia is a really complex idea, as well as very very old. I mean thousands of years old. It changed throughout time, there were many nuances etc, and I don't want to go into too much detail, as your question is not about Tianxia. If you're interested, I recommend reading China and International Relations, by Zheng Yongnian.
While I don't want to go in to too much detail on Tianxia, there are a couple of things you need to know:
The idea of Tianxia was able to hold for millennia because it made sense up to that point. Even if China was conquered (fe by the Mongols), they could easily claim the Mandate of Heaven, which gave them the right to rule. This changed when the Europeans came.
Now, at first the Europeans were quite peaceful. They send trade envoys, asked for permanent representation in the capital, and while they weren't glad with trade restrictions, they mostly adhered to them (except smugglers of course). When Britain started to export massive amounts of opium from India to China (they had a trade deficit they wanted to reduce) and the emperor said no that the first Opium War broke out. This is of course a simplification of what truly happened, but it's not about the first Opium War in 1839.
Anyway, this was just the first of several wars and conflicts with Western Powers, all resulted in the humiliating loss of the Chinese. The next 110 years are known as the Century of Humiliation. There were wars with Japan, wars with the Europeans, rebellions and uprisings etc. Generally, not a good time to be an emperor.
China had known chaos before. Plenty of dynasties were brought done by foreign invaders, by rebels and by traitors, but always the winners would have claimed the Mandate of Heaven. Europeans tend not to do that. So you had chaos in China, clearly indicating that the Qing dynasty didn't have the mandate of heaven, but no one was claiming it. When the Qing fell, in 1911, there were some contenders for the throne, but none achieve dominion. There was a Warlord Period, later the communists fought with the nationalists in a civil war (and many other wars as well), but the point is that no one claimed the Mandate. Which means the Tianxia was finally brought down. The Chinese started to look for another explanation and sovereignty was right there. Many Chinese had gone to Japan to learn 'Western teachings' as western medicine. Sun Yat Sen (original leader of the nationalists) had actually gone to western countries and was fluent in English. More and more Chinese either went to Japan to study western teaching, or went to schools in China that offered those teachings. Leaders saw in sovereignty also a good reason why the attacks on China by foreign powers were wrong ('they violated our sovereignty!') and used it to rally supporters.
I need to go and cook now, will return later if you have any questions!
First of all remember that the concept of the nation-state came from the West. Therefore the development of nation-states (or nationalism itself for that matter) in Asia were either an effect of European colonialism, or were created as a sort of defense against colonialism. For example, Japan began it's path towards a nation-state during the Meiji Restoration in the 1860s. The idea (in part) was to adopt Western styles of consolidation and industrialization in order to better compete with and defend themselves from West.
In the book Imagined Communities, Benedict Anderson argues that colonialism created a sense of "nationness" between its previously disconnected victims by: --the building of infrastructure and insertion of modern transportation meant to ease the extraction of resources (breaks down isolation of natives from the rest of their "nation") --Forcing Assimilation in order to create a larger network of bureaucrats needed to administer colonies --spreading Western Education to natives (especially in the French case in Indochina where the ideas of the French Revolution were allegedly taught)
Anderson uses the example of Indonesia (Originally the Dutch East Indies), which prior to European incursion was a series of fairly isolated islands with people of different languages, beliefs, etc. Centuries of Dutch rule ironically gave the people of these islands to connect with one another, and even led to the creation of the Indonesian language, a mishmash of several native languages and Dutch.
As for China specifically, I would again trace the development of the concept of the nation-state with the incursions of European colonizers. Revolutionary leaders like Sun Yat-sen (who studied in the United States) knew that the only way China could survive against the Western Imperialists was to adopt a western style of governance, of nationalism, and of industrialization.
While others have pointed out that the concept of the nation-state is indeed a European state, the rise of a nation-state in Mongolia is very much linked to historical events in China during the early part of the 20th century.
Mongolia did not fare very well under Manchu rule in the Qing dynasty, severely impoverishing the region. Furthermore, in the 10 years leading up to 1911, the Qing administration changed its policy of forbidding Chinese (Han) settlement and interaction with Mongols, and begin to implement policies designed to assimilate and ultimately destroy independent Mongolian identity through Chinese settlement. This became known and growing resistance to the local representative of the Qing, Sando, precipitated the move towards full-blown independence.
In 1911 prominant nobles and buddhist leaders met with the future Богд хаан, the 3rd highest leader of Tibetan buddhism and the spiritual leader of buddhism in Mongolia. A plot was hatched to send a delegation in secret to Russia to ask for support in a push for independence. The Russians minimally agreed, but the Qing were informed of the delegation and attempted to crack down in Mongolia. However growing unrest in the rest of China basically gave the Mongol leaders confidence to act decisively. The Qing governor (viceroy technically) Sando had little choice, having only 150 soldiers, and on 30th November 1911 a Temporary Government was established. This was followed by a Provisional government, and then the official installment of the Bogd Khaan as emperor of the new Mongolian state at the end of December 1911.
Effectively this saw the emergence of a modern Mongolian state, even though it was very much an early 20th century theocratic monarchy. After what one might term a very extended 'medieval' period, Mongolia emerged as a distinct region again with a distinct and coherent socio-political identity. That identity was formed against the eclipse and failure of the Qing dynasty in light of events in China, as well as the broader political engagement of Russia at the time in both Asian and European politics. It also indicated the general alignment of Mongolia towards Russia rather than China.
The state itself fused remnants of Qing rule, with Mongolian buddhist theocracy, and early 20th century Western political ideas. A flag was adopted, as well as a parliament, and a government with different ministries.
However Mongolia also had great difficulty in getting its independence 'recognised'. The newly emergent Republic of China still considered Mongolia a part of its territory. And Russia's position was that Mongolia was an autonomous region within China. Being sandwiched by these two states, Mongolian diplomatic attempts to contact, let alone gain recognition, by other foreign powers, let with very little success. Continued pressure from China, and continuing reluctance from Russia to recognise true independence, alongside the focus-drawing power of WW1 and then the Bolshevik revolution let Mongolia increasingly vulnerable to China's moves to bring Mongolia back into their fold.
The Bogd Khaan ruled until 1919, at which time Mongolia underwent a period of brief subjection to Chinese rule again, until the People's Revolution of 1921 saw again the emergence and establishment of an independent Mongolian state, this time along Communist lines. But that might be a story for another day.