They call it Byzantine because it's common historiographical term and it is how historians used to call this state. For example, Muscovy is in reality called Grand Principality of Moscow.
Also we can use "Byzantine even didnt have a Rome" argument (or loss of Rome for 6 centuries still makes them Romans?).
The Western Roman Empire had its capital at Mediolanum, not Rome. Plus, Constantine changed the capital to Constantinople anyways, meaning they held on to the Roman capital until 1453.
So make the Byzantine Empire called Rome and Trebizond be a separate empire with ambitions to retake Rome?
Also, the Byzantines weren't considered Romans by the Western nations, which is part of what the tag names indicate. Being the Roman Empire indicates general acknowledgement of being the Empire itself. At least, that's what I got from this whole affair. Sure, the Eastern Romans called themselves the only Rome but, without implicit and engrained acknowledgement of that fact (which was removed IRL due to the Pope and Charlesmagne and the Empress, stuff happened, yup), the only way to enforce that is militarily, which is what the territorial requirements is representative of.
Likewise, a successor to Rome can be an entirely different empire with weak claims as long as they can enforce them. The idea of Rome went from being an empire to becoming THE empire. Russia, the Ottomans, Italy, Romania, etc. If anyone was actually strong enough to conquer all of old Rome's territory, I doubt there would've been much opposition to that particular nation labelling itself "Rome"
The Byzantines were not only a continuation, but they were recognized by everything from Bulgaria to China. Eastern Europe, the Turks and Muslims, and all the way to the Ming (who didn't even know that Byzantium had lost Antioch, it had been so long since the two countries had met) recognized it as the Roman Empire. Both the Russians and the Ottomans built their claims not on territory, but on the fact that each of them had legitimate claim to the Byzantine throne. While the Ottomans would be the official inheritors, having been crowned by the Patriarch, having intermarried many times with Byzantines, and declaring themselves officially the Roman Empire upon taking Constantinople, they would hold the title by any given succession laws. Russia would be a strong claimant, but the Ottomans would've been the direct heirs.
The whole reason the Ottomans did what they did was to reform the Roman Empire under Islam.
Sure Byzans inherited lands of east Rome, but it was not the same as the idea of Rome. Byzantium was a hellenistic orthodox christian empire, a big change away from Rome. Rome as a "Roman empire" had disappared due to Christianity as it started excluding citizens cause of their religion
If I were to ask any Roman emperor what his nation is, he'd probably just follow up with a question of "What is this 'nation' you speak of." As national identity has no meaning in those times and borders, well Rome as the only light in a world of darkness has no borders. Its just Rome and the uncivilized world outside it. Rome is not Christian, its not Greek, it takes whatever is beneficial and discards the rest, it is without precedent on what people are true Romans or what truly is Rome. Rome is the idea of an Empire incorporating all the lesser cultures under the banner of civilization.
Because the Romans never denied anyone citizenship based on certain habits.
National identity was practically everything back then. Heck, it was proto-racism. The Romans hated everyone who wasn't a Latin, and declared them barbarians. They set out to conquer these barbarians and refused to grant them equal citizenship. What the Romans would've defined as being Roman is, basically, anyone born in Rome who speaks Latin. Later, they expanded their citizenship to all of Italy. Greeks and Egyptians came to be accepted. The Celts, Germans, and Spaniards were the last to see any sort of benefit from the Roman Empire, and they got it mostly just because Rome became Christian. The "Idea of Rome" that you speak of, the one you say is without precedent, is an idea shared by many other empires of the era. China had the exact same idea. Alexander had an idea pretty close. The Persians felt this way towards their nomadic neighbors. There is a very clear precedent for the way Rome behaved, and the idea of empire was hardly a Roman innovation. Empires are, by definition, multiethnic. Here:
a group of nations or peoples ruled over by an emperor, empress, or other powerful sovereign or government: usually a territory of greater extent than a kingdom, as the former British Empire, French Empire, Russian Empire, Byzantine Empire, or Roman Empire.
Would you deny that the Roman Empire, as a country, simply ceased to exist entirely upon Constantine's conversion. Did the throne simply disappear, the authority dissolved? Did Constantine's conversion alone, with not a single other factor, mean that all of Italy and Greece were left without any leadership at all for centuries?
The fact that the Romans enforced Latin customs across their empire should be proof enough that they had a defined identity. The Gauls were made to the French. The Celtiberians and the Basques became the Spaniards (Except for those in modern Basque Country). Dacians became Romanians. Illyrians became the Albanians (light Latin influences) and the Dalmatians (heavy Latin influences).
The Romans slaughtered Celts en masse, burned down entire cities, expelled the Jews from their homeland, and committed outright genocide against Carthaginians (Phoenicians in Africa). It wasn't until much much later that the Romans became a pan-ethnic empire, although by that point it was largely full of Latinized peoples. Those who remained were tolerated, and the Byzantines continued this better than anyone else.
If you asked a Roman emperor what his nation was, he would say thus:
"My nation is Rome, my people the Latins. We are the sons of Mars, bred for war against the barbarians that surround us. Celts, Germans, Iberians, Numidians, Persians, they all defy civilization. I live my life by Roman law, I heed advice from the consuls, and I know my heritage as any Roman should. I do not let the barbarians stain me. Not even their food shall enter the gates of Rome, on penalty of death. My nation is Rome, and I am a Roman."
Now, here's a direct quote from Constantine XI Palaiologos:
I know the countless hordes of the impious will advance against us, according to their custom, violently, confidently and with great courage and force in order to overwhelm and wear out our few defenders with hardship. They attempt to frighten us with loud yells and innumerable battle cries. But you are all familiar with their chattering and I need say no more about it. For a long time they will continue so and will also release over us countless rocks, all sorts of arrows and missiles, like the sand of the sea. But I hope that such things will not harm us; I see, greatly rejoice, and nourish with hopes in my mind that even if we are few, you are all experienced and seasoned warriors- courageous, brave, and well prepared. Protect your heads with shields in combat and battle. Keep your right hand, armed with the sword, extended in front of you at all times. Your helmets, breastplates and suits of armor are fully sufficient together with your other weapons and will prove very effective in battle. Our enemies have no and use no such weapons. You are protected inside the walls, while they will advance without cover and with toil.
For these reasons, my fellow soldiers, prepare yourselves, be firm, and remain valiant, for the pity of God, Take your example from the few elephants of the Carthaginians and how they dispersed the numerous cavalry of the Romans with their noise and appearance. If one dumb beast put another to flight, we, the masters of horses and animals, can surely even do better against our advancing enemies, since they are dumb animals, worse even than pigs. Present your shield, swords, arrows, and spears to them, imagining that you are a hunting party after wild boars, so that the impious may learn that they are dealing not with dumb animals but with their lords and masters, the descendants of the Greeks and the Romans.
