It seems that in the study of Chinese history (at least in the popular sense), all the Chinese dynasties have received their own special appellation, almost as if they were separate empires, even though Chinese are not unique in having dynastic rule. In contrast to the study of other ancient empires, dynasties receive much less specific attention, and I would argue, for example, the Julio-Claudian or the Flavian dynasties of the Roman empire is much less well known than the Han, or the Tang dynasties to the average educated person. I was wondering the reason for why Chinese dynasties are treated somewhat differently from the dynasties of other empires.
Firstly, there's nothing necessarily preventing you from still calling all the dynasties 'China', in the same way that saying 'France' could mean the Kingdom of France (Merovingian, Carolingian, Direct Capetian, Valois, Bourbon and Orléanist), two French Empires (those of Napoleon I and Napoleon III) and five French Republics.
Secondly, to add to what /u/lcnielsen has said, it's not that we treat them differently, but that they treated themselves that way. The names of dynasties do not line up with the family names of the families ruling them. The Mongol Yuan emperors had no surnames; the Han Chinese Ming emperors were the Zhu family; and the Manchu Qing emperors were all members of the Aisin Gioro clan. The terms Da Yuan, Da Ming and Da Qing were all voluntary appellations. Now, there were various reasons for this, but put simply these were empires calling themselves the '(Great) [whatever]'. The use of 中國 Zhōngguó interchangeably with the dynastic name was largely a Qing Dynasty-era innovation, whereas previously it referred only to a particular portion of the Chinese heartland.
Sorry, I just don't see how this is a peculiarly Chinese thing. There are many cases where a dynastic name or ethnonym is common or even preferred usage compared to a geographic name. In picking the early Roman Empire you're picking a peculiar case (where there is a lot of continuity between dynastic changes) where, nevertheless, the dynasties are quite commonly referred to. This is even more true for the Byzantines (e.g. the Macedonian Dynasty).
In Europe off the top of my head:
If I were to give realms in the Middle East I don't think I would get done today; using dynastic or similar names is more or less the rule there, at very least in the Islamic era.
In China, it's also the case that dynasties give a natural method of periodization. That's not least because a dynastic change in China wasn't just a quiet change of the ruling family, but occurred with rebellions, systemic collapse, invasions, etc, many of which took incredible casualties (An Lushan Rebellion, Ming Collapse, etc). There just isn't the kind of administrative continuity you see in e.g. the Roman Empire, in spite of the fact that there were obviously many persistent methods of administration, bureaucratic institutions, and cultural norms. There isn't really a continuous country called "China" that just changed ruling dynasties a couple of times, so it makes far more sense to speak of dynasties when any level of detail is desireable.
The main reason why we do this is because the Chinese sources themselves do so. In Chinese tradition, a change in ruling clan meant the end of one state and the creation of a new one, with a new name. The Liu clan established the Han state, the Cao clan ended Han and established Wei, and the Sima clan ended Wei and established Jin. In practice there were frequently several different clans claiming to be the imperial clan, but in ideological terms the pattern was held. In contrast the Roman Empire went through rapid changes in "dynasty" with no real impact on the Roman state's self-image.
A secondary cause is that, as a result of this view of the state that was very focused on the ruling family, official court histories which are among the most important sources in Chinese historiography are, with very few exceptions, compiled among dynastic lines (i.e. The Book of Jin, The Book of Northern Qi). Out of dynastic-focused historiography grew a view of Chinese history as a cycle of the rise and decline of dynasties, with vigorous and capable founding emperors giving way to lesser descendants until the cycle is restarted by a new heroic founder.
The are more practical reasons to look at Chinese history on a dynastic basis. For one, Chinese territory, population, institutions and foreign relations changed significantly between Qin and Qing, and the major dynasties are manageable chunks of time for looking at these developments. Changes over the course of a dynasty could be just as important as the changes between dynasties, but we're still able to broadly compare, say, Tang literature versus Song literature, or the Ming military and the Qing military. During the several periods of division, it's also helpful to be able to distinguish between the multiple coexisting states which ruled only part of China but claimed all of it.