Recall that Imperial China cannot be considered as one contiguous package; individual dynasties had varying attitudes to war ranging from the expansionist and heavily militarised Tang (618-907) to the more diplomatic Song (960-1279). That being said, early Imperial China most certainly had an aristocracy that for a period constituted the core armed forces. Aristocrats in imperial Chinese society rose to prominence during the Han dynasty (206BC-220AD), gaining significance following the death of Han Emperor Ling (r.168-189).

Ling's control of the throne prior to his death had been heavily influenced by the eunuch faction, which had not only reached near ubiquity in the Han court by this point but had vested interests in essentially all aspects of governance, the empire, and imperial intrigue. This period of eunuch domination would be swiftly curtailed with Ling's death, resulting in a massacre of eunuchs by military officers, which shifted the balance of power in the court entirely in favour of what by now was an establishing aristocracy. The Han dynasty was heavily destabilised by these events and would never recover, eventually resulting in the division of the state by rival warlords [in a period known as the Three Kingdoms (220-280AD)].

Briefly reunited under the Jin dynasty (265-420), imperial China would next see long-standing unification under the Tang dynasty in 618AD. However, the aristocracy which developed in the late Han period during a collapse in imperial authority continued to develop during this time. Tang historians would make note of this trend in the pervasiveness of the aristocracy in the 9th century, and it is during this dynasty that Chinese aristocracy would reach both their peak in military affairs and in fact their downfall.

By the time of the mid-Tang dynasty it was relatively easy for several surnames and lineages to be traced as key long-term constituents of a sort of aristocracy. Whilst these families were not officially recognised as such by the emperor, they enjoyed de facto recognition with wide-ranging privileges including the right to entry-level bureaucratic postings and much legal exemption, often sought-after. Many of the more famous families enjoyed empire-wide recognition, with considerably more prestige attached than to fleeting dynastic families which appeared between the Han and Tang, and often even to the Tang imperial line of Li. These families were both officially and unofficially ranked in terms of prestige - the latter of which rankings even placed the famous four families of Hebei above that of the dynasty itself (Li). Many of this elite even considered the imperial family as tainted by the blood of northern outsiders.

The importance of this aristocracy at this point in Tang history cannot be understated, and their role in military affairs was particularly crucial. The Tang Military System incorporated a hereditary Northern army, an aristocratic central army and a regimental army (fu bing), composed of the sons of elite Northwest families (then considered the most politically important region of China). The regimental army was first introduced in the Southern & Northern dynasties period (420-589) by a group of both non-Chinese and Chinese elites who actively cultivated military tradition. It was this system the Tang emulated when they refounded the regimental army in 636. As stated prior, men were sourced from wealthy landed families capable of permitting adult males of their family to serve exclusively in the armed forces - as opposed to in many less-wealthy families where adult males were required for agriculture. Regimental forces recruited this way enjoyed many privileges; including exemption from tax and labour service [including the dreaded corvée], and an allotment of land to work. Critically these soldiers were expected to provide basic provisions for themselves - including a weapon - whilst armour and more specialised weaponry was state-provided. This system ensured state spending on what became an exceptionally large army was minimised, and these conscripted troops were utilised variously amongst police services, for service in expeditionary armies abroad and to man the frontiers.

The Northern army was almost exclusively composed of descendants of the founding Emperor Gaozu's (r.618-626) original army, whilst the central army was recruited from sons of elite families both of the immediate area inside and around Chang'an and those of the Northern army. These armies constituted the core Tang military force for much of the dynasty, with troops variously serving alongside mercenaries [often Turkic] in all types of military operation. This system pervaded all aspects of the early to mid Tang court; the ruling elites of the Tang [and prior Sui dynasty (589-618)] were steeped in military tradition. However, it began to break down in the latter half of Tang rule. Growing threats at the frontier with Tibet, Nanzhao, Turkic and other states required long-term service of armies designed to man the frontiers against invasion. These armies were essentially exiled to the frontier for indefinite periods, immediately reducing the attractiveness of service to the aristocracy. Draft-dodging became commonplace amongst the elite, whereas previously the military system had been seen by the elite as a method of cultivating experience for later service within the empire. Regimental armies thus had to increasingly rely on drafts from poorer households, the prestige of service sharply declined, and by 749 the regimental army was all but abolished.

This emphasis on semi-permanent frontier armies marked a significant shift in Tang attitudes to warfare, and, along with other political shifts in the period - notably the crisis of the An Lushan rebellion (755-763), resulting in the decline in authority, prestige and importance of the Chinese medieval aristocracy and its essential dismantlement by the Five Dynasties period (907-960). Whilst aristocracy of some form would resurge much later in the Qing dynasty (1644-1912), it would neither take the same form nor have the same importance in military affairs as had been the case in the Tang and several other dynasties of antiquity.

Your second question can be argued variously. Study of Chinese history is still very much in development even domestically within China. The vast, often complex history of China isn't covered much if at all in Western education - whereas great attention is granted to empires such as those of Greece [Alexander] and Rome. Whilst there is little awareness of Chinese history amongst the general public outside of the region, to argue this is the sole reason would be disingenuous. Aristocratic military affairs in China were not characterised by massively disparate fighting styles or attitudes, it is thus difficult to define the role of the military aristocracy as iconic in the same way as can be done with several of the examples you mentioned. A well-to-do individual who provides his own arms and serves the Tang state for the prestige associated does not capture the same image as a hopilite or knight does.

Sources:

China: A History - John Keay

China's Cosmopolitan Empire: The Tang Dynasty - Mark Richard Lewis

I'm afraid your perceptions of China are wrong. China had a vast military aristocracy on a scale that rivaled almost any pre-modern society. The last dynasty of China (the Qing) was ruled by a minority group from the Northeast called the Manchu. These Manchu were organized into 8 banners. After conquering the Ming the 8 banners were a hereditary military class that occupied key parts of many big cities around the Empire.

This military aristocracy worked for a time, but fell into economic hard times (at least the common Manchu garrisons) and couldn't overcome several major Rebellions in the 19th Century. By the time they were disbanded many of the Han majority hated them and there were several massacres of Manchu garrisons around China. (See Edward Rhoads Manchu and Hans:Ethnic Relations and Political Power in the late Qing and Early Republican China for more detailed accounts)

I think the reason many don't think about Chinese military aristocracy is a mix or Euro-centric focus in Western history as well as contempt for what happened within older Empires such as the Ottomans or China in the Modern Era. These are just my conjectures, as to why they are not as "iconic" but there could be other reasons as well.

Adding on to the other posts, it's not actually the case that Song China (for example) was alone in being a civilian-led society.

I'll use one example from Korea (see The Origins of the Choson Dynasty for the information). The Goryeo dynasty, which ruled the entire peninsula for most of the time between 936 and 1392, began as a confederate regime of provincial strongmen holding military power.^1 This changed significantly in the later 10th century, when two kings weakened the authority of local strongmen and reorganized the government.

What's more relevant to your question is that the military branch of the reorganized Goryeo government was undeniably inferior to the civil branch before the military coup d'etat in 1170. The highest civil official was the Chancellor, who held a position within the junior first grade of government. By contrast, the highest military official, the Supreme General, was a senior third grade post. Military affairs were coordinated by the civilian-dominated institution called the Security Council, and even in wartime high-ranking civil ministers would often be given supreme field command; Gang Gamchan was a civilian. So the early Goryeo ruling class was dominated by civil elements, namely the great descent groups.

In the earliest years of Goryeo it had had irredentist aspirations for what is now northern Korea and southern Manchuria, but soon Goryeo shifted into a state that generally sought to maintain the status quo and it had few means of successfully disrupting it anyhow. So a state of war was very much not the norm, unlike their northern contemporaries the Khitans. It was quite a different situation from, for example, the Aztecs that you mentioned in OP (any questions about Aztec warfare would be better answered by /u/400-rabbits though).

Not only that, remember that chiefdoms and states aren't the only societies around. Most non-agricultural societies do not have a ruling class or aristocracy at all, although of course there are exceptions. You shouldn't conflate states with societies - 1500 China and 1500 Moriori are both very much societies, but only one is a state.

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^1 I should note that there's still some discussion about this.

The closest thing I can think of for a military aristocracy in China is /府兵/军户 during Early Tang and Ming Dynasty. Who are giving land in return for providing at least 1 combatant during mobilization. (Hence why Mulan [real or legend] need to serve in place of his father, otherwise all her family's property will be confiscated) However, these soldiers-farmer are living too close to poverty to be really considered an aristocracy.

In other times, there is no dedicated military class in China. For Qin/Early Han, it's pretty much full mobilization. For Later Han, Song, Non-Manchu population of Qing, it's essentially career soldiering and such career to consider to be a low class occupation.

As for elite units within the Chinese army, they exists. But I guess there isn't enough movies made about them for them to be iconic.

Jingyiwei of the Ming Dynasty but they had a similar negative perception from the public (as the late/past-warring states samurai) and were abusive with their power. They were given their power for the purpose of stamping out dissent, officials disloyal to the emperor, and for reinforcing hardline, conservative Ming values and laws.

Maybe they weren't as iconic because of fewer Western historiography of pre-modern China? Maybe Japan relatively had more or better writing of their history into European languages.