There are remarkable parallels between the two, it could be argued. There is something of a myth of Chinese continuity which isn't really warranted at face value when you examine the history more closely. Likewise, there is another myth that one day there was a Roman Empire, and the next day Romulus Augustus was deposed and suddenly Europe was cloaked in darkness, and the legacy of Rome extinguished completely. Real events are considerably more muddy and don't necessarily conform nicely to narratives.

The Tang dynasty, like the Roman Empire, splintered in internal divisions and revolts, leaving behind a rump state that preserved elements of the previous structure, as well as a plethora of semi-"barbarian" successor states controlling portions of the former extents that had their own synergies of "Chinese" and non-Han properties, which one can squint at and see similar reflections of in the Germanic Roman successor states and the Eastern/Byzantine Empire. When Genghis rode into the gates of Zhongdu (Beijing), he had driven out the Jin, who were originally Jurchens from Manchuria, who had in turn displaced the Liao dynasty, who where originally Khitans from Mongolia, who had in their turn taken over when the Tang dynasty fell to pieces. Then his grandson completed the conquest and wiped out the Song dynasty that had remained in control of the southern portion of the old empire. Imagine Charlemagne being coronated Holy Roman Emperor in Rome, and then rather than his grandsons dividing his empire between them in civil war, one had proved powerful enough to maintain control, and then turned and captured Byzantium and the rest of the former Eastern Empire.