Using Latin as primary means of communication, putting emphasis of Rome as actual city, deriving legitimacy of Emperor from popular mandate are three ideas of pre-division Rome which were preserved in Charlemagne's empire, and they weren't as prominent in ERE.
I have absolutely no clue where you see the latter two in Charlemagne's empire. "putting emphasis of Rome as actual city"[sic] - what does that even mean? While Rome was technically part of Charlemagne's empire, it was a peripheral backwater, important only because the Pope resided there. And Frankish emperor's weren't even remotely legitimized by a popular mandate - they were legitimized by the Church, which paid no heed to the opinions of commoners.
The ERE also technically controlled Rome in the same manner as Charlemagne, and unlike in the West, Eastern Emperors could still be acclaimed by the military and people of the capital.
Wasn't Latin official language up until Heraclius?
Exactly. And Heraclius became emperor before the Slavs or the Arabs were a major threat to the ERE.
That doesn't make Ottoman Empire Roman, and no, the "subjects" changed from the times when ERE controlled Anatolia by inclusion of ethnic Turks.
Except the ERE also ruled over a large Turkish population for centuries and itself presided over a number of large demographic shifts during the millennium for which it existed.
Primary means of communication was ancestral language to modern Turkish and the actual goverment language was Osman - mashup of Persian, Arabic and Turkish, so 2/3 of it's consistent elements were not related to Latin at all.
If the identity of an empire is determined by the actual native language of most of its inhabitants, then the Roman Empire ceased being "Roman" long before Constantine split it in twain. Afaik Latin was only ever the native language of a large plurality of Romans (at its peak, maybe around 50% of the population) and not the majority. Inside the empire, the actual lingua franca from a very early age was Koine Greek.
These arguments reflect a long tradition of western Orientalism which seeks to monopolize the mythology of the SPQR by excluding certain groups from both our present identity as well as our mythology. It's the reason why Gibbons considers the HRE but not the ERE to be Roman; why the Orthodox sphere instead legitimizes the ERE and Russia; and why both deliberately exlude, other and orientalize Muslim states that claimed legitimacy from the SPQR - most notably the Ottomans. These arguments are not sound in themselves, but are contingent on Western political theology.
By contrast, Chinese nationalist mythology is more interested in assimilating the histories, identities and traditions of peoples within its sphere of influence. Hence the concept of "dynasties", which is, and always has been, a propaganda tool to legitimize political interests and delegitimize independence movements in the wider Sinitic sphere. Broadly, it's used to attack Mongolian, Uyghur, Manchu and Tibetan identity, and more narrowly, to establish the idea of "China" as a monolithic historical entity with some inherent claim to political power. It's the Mandate of Heaven transfered to a secular belief system.
And just like the idea that the gods ordained some guy to rule with absolute power, we should understand that both these mythologies are ultimately reductive and stand in the way of a complete understanding of history. Namely that monolithic empires that exist for thousands of years don't exist. I.e. the Han Empire collapsed in the 3rd century, and it would only be co-opted as part of a "Chinese" identity centuries later.
