An important thing to note about emperors of China is that we almost never refer to them by their 'actual' original names except when referring to the time before they become emperor, as the use of the characters in an emperor's given name became taboo during their reign, and indeed for the purposes of their reign itself, they would simply be 'the emperor'. Specific names for the emperors themselves were conferred posthumously, and different standards have existed. Let's take some examples through time:

Emperor Wu (武) of Han (漢), or Han Wudi (漢武帝), the seventh emperor of the dynasty (r. 141-87 BC), was in fact named Liu Che (劉徹). The name 'Wu' was a sort of retrospective title (in this case, meaning 'martial') conferred posthumously, and indeed it is a shortening of a fuller posthumous name, Xiao Wu Huangdi (孝武皇帝; 'filial and martial emperor'). Theoretically, we could also refer to him by his temple name of Shizong (世宗), prefaced with the dynasty name, Han, to form Han Shizong.

Emperor Taizong (太宗) of Tang (唐), or Tang Taizong, (唐太宗), the dynasty's second emperor (r. AD 626-649), was originally named Li Shimin (李世民). 'Taizong' was his temple name, again conferred posthumously. The short form of his official posthumous title was Wen Huangdi (文皇帝; 'civil/literary/cultured emperor'), a shortening of Wen Wu Da-Sheng Da-Guang Xiao Huangdi (文武大聖大廣孝皇帝; 'civil, martial, sacred, broad, filial emperor'), and so we could theoretically call him Emperor Wen of Tang or Tang Wendi.

Toghon Temür, the last emperor of the Mongol Yuan (元) Dynasty to rule China proper (from 1333-1368, though he ruled the Yuan rump state in Mongolia till his death in 1670), is often referred to by his Mongol name, but he also held Chinese titles. The Ming referred to him as Emperor Shun (順), but one could theoretically use his temple name of Huizong (惠宗), so he could also be known as Emperor Shun of Yuan/ Yuan Shundi, or Yuan Huizong.

The fourteenth emperor of the Ming (明) Dynasty, commonly known as the Wanli (萬曆; 'myriad calendar') Emperor (r. 1572-1620), was born Zhu Yijun (朱翊鈞). His short posthumous name was Xian (顯; 'prominent', 'distinguished'), short for Fantian Hedao Zhesu Dunjian Guangwen Zhangwu Anren Zhixiao Xian Huangdi (範天合道哲肅敦簡光文章武安仁止孝顯皇帝; I'm not even going to try); his temple name was Shenzong (神宗). If we so wished we could therefore call him Emperor Xian or Ming Shenzong. However, from the first Ming emperor* onwards, emperors began using a single era title through their entire reigns, instead of adopting new era titles on regular occasions (Han Wudi, for instance, went through 11 eras in his 57-year reign), which means that it becomes viable to refer to an emperor as the '[Era Name] Emperor' as a short of shorthand for 'the emperor during the [Era Name] era'.

And this sort of convention was retained by the Qing (清). The fifth Qing emperor, commonly known as the Qianlong (乾隆; 'imperial/masculine [and] prosperous') Emperor (r. 1735-1796/99), was born Hung Li/Hongli (ᡥᡠᠩ ᠯᡳ/弘曆) of the Aisin Gioro (ᠠᡳᠰᡳᠨ ᡤᡳᠣᡵᠣ) clan. It is a crying shame that we do not refer to him as the Abkai Wehiyehe (ᠠᠪᡴᠠᡳ ᠸᡝᡥᡳᠶᡝᡥᡝ; 'supported by heaven') Emperor, but you can't have everything I suppose. Theoretically, we could call him Emperor Chun (純; 'pure')**, or even better use his Manchu posthumous name, Emperor Yongkiyaha (ᠶᠣᠩᡴᡳᠶᠠᡥᠠ, 'perfect/complete'); or we could use his temple name and call him Qing Gaozong (高宗; G'aodzung (ᡬᠠᠣᡯᡠᠩ) in Manchu). However, as with the Ming emperors, as his entire era was known only by a single title, we mostly call him the Qianlong Emperor instead, though some do use Hungli/Hongli or Gaozong to refer to the man himself.

In some ways this spiel was perhaps a little superfluous and only really goes to affirm the premises of the question: that at some point we move from using names that refer directly to the emperors themselves, irrespective of whether they were names used during their lives (short posthumous and temple names), towards preferring the style of '[Era Name] Emperor'. The practical reason for it being possible I have mentioned above: the Ming emperors used only a single era name throughout their reigns.*** But the short and simple explanation for why we use era names over temple or posthumous names for the Ming and Qing more or less boils down to... because we do. As you'll note from before, for earlier emperors such as the Han, we go for posthumous over temple names, but from the Tang we're looking at temple names instead. Part of the reason for this, as suggested by Dubs (1945), is that posthumous names underwent significant bloat, as has been illustrated above: between the Hang and Sui dynasties, emperors' full posthumous names, excluding the 'emperor' portion, were typically only two characters, while Tang emperors used around seven, making the two-character temple name much more practicable. However, the cause of the transition to using era names is somewhat unclear, and historically the temple name form was also sometimes used during the dynasties themselves. At best, we can say how it was able to happen, but the why is less so.

* Real name Zhu Yuanzhang (朱元璋), reign title Hongwu (洪武; 'flood martial'), temple name Taizu (太祖), short posthumous name Gao (高; 'high').

** short for Fatian Longyun Zhicheng Xianjue Tiyuan Liji Fuwen Fenwu Qinming Xiaoci Shensheng Chun Huangdi (法天隆運至誠先覺體元立極敷文奮武欽明孝慈神聖純皇帝)

*** except for Zhu Qizhen (朱祁鎮), temple name Yingzong (英宗), short posthumous name Rui (睿; 'shrewd'), who is typically referred to as the Emperor Yingzong because he reigned twice, once as the Zhengtong (正統) Emperor from 1435 to 1449, and again as the Tianshun (天順) Emperor from 1547 to 1564.