The difference here is between temple names and era names. A temple name, bestowed posthumously, was considered an actual name of the emperor in question. As such, you have Li Shimin as Emperor Taizong of the Tang, and Hung Taiji as Emperor Taizong of the Qing. An era name, on the other hand, referred to a particular period of time, not the emperors themselves. Up until the Ming Dynasty, most emperors typically ruled across several named eras, usually between four and nine years in length, unless of course they died before they began a second era. From the Ming onward, however, emperors almost invariably chose an era name at the start of their reign which would remain in place for its entirety (hence sometimes they are referred to as 'reign names' or 'reign titles' during the Ming and Qing periods.) Consequently, the construction '[reign name] + Emperor' became valid to use in retrospect, and indeed would also come to become standard practice during the emperors' reigns. The crucial thing here is that the reign name refers to the reign itself, not to the person of the emperor. Thus they are the emperor of that particular reign period, not an emperor by that name.

So, to take your two examples, Li Shimin named (unusually) the entire period of his reign the Zhenguan era, and was posthumously known as Emperor Tang Taizong. Zhu Di named the period of his reign the Yongle era, and was posthumously known both as Emperor Ming Taizong and Ming Chengzu. However, Li Shimin was not Emperor Zhenguan, nor was Zhu Di emperor Yongle, because (I seem to keep repeating myself here) Zhenguan and Yongle were the periods of time during which these emperors reigned, not the emperors themselves. Of course, partly out of laziness and partly out of convention you do often find people calling Ming and Qing emperors 'Kangxi' or 'Qianlong' or 'Wanli', but from a strictly pedantic perspective this is somewhat incorrect.

Ming and Qing emperors only had one reign name/era title 年號, while previously emperors could have many. Tang Gaozong for instance had 14. Tang emperors are usually known by their temple names, which were given posthumously. Tang Taizong’s personal name was Li Shimin 李世民, and his era title was Zhenguan 貞觀. If he was known by his era title, he would likely be called the Zhenguan emperor in English. Temple names are often quite generic, so one needs to specify the dynasty when calling them by their temple names. Gaozu 高祖 for instance is a temple name commonly given to the founder of a dynasty (another one is 太祖), so one would need to specify Tang Gaozu to avoid confusion with other dynastic founders (e.g. Emperor Gaozu of Han, aka Liu Bang 劉邦). Thus one can’t really write “the Gaozu Emperor”, unless it’s clear in context which Gaozu we’re talking about. I believe that’s where the convention come from.

Ming and Qing emperors had temple names too, but it is convenient to refer to their era titles as they are less easily confused and there’s only one for their whole reign. But the era title is just that: the name of the era. It’s named for the emperor, but refers to the period of time he (or very rarely she) was on the throne. So technically the Yongle Emperor is the emperor of the Yongle era, if that makes sense, thus the definite article.

In Chinese both temple names and era titles are commonly used, so you will see people use both the Yongle Emperor 永樂皇帝 and his temple name Ming Chengzu 明成祖. Note that in Chinese, “Emperor” is present when using the era title Yongle, but only the dynasty (Ming) is used when using his temple name. This isn’t systematic though, as it’s uncommon in Chinese to refer to him by his personal name 朱棣, but it’s common to refer to Tang Taizong by his personal name Li Shimin. Not sure why.

For the Qing, they have long personal names in Chinese that come from Manchu, so they are only ever referred to by their reign titles in Chinese, a practice that is followed in English.