Legally speaking, Byzantium was Rome--not Eastern Rome but just Rome, all of it, since Odoacer returned the Western regalia to Zeno.
On the other hand, historians aren't lawyers. And shouldn't be.
The seventh-century crisis was fundamentally transformative for Rome/Byzantium. Take the economy, for instance. The Principate had been an empire of cities; and while city-state autonomy was curbed under the Dominate, the sixth-century empire remained a highly urban civilization. Then, in the seventh century, the Byzantine urban economy collapsed entirely. Constantinople's population fell from half a million under Justinian to a quarter of a million under Heraclius, and from that to a mere 40,000 under the early Isaurians. When the city did recover under the late Isaurians and Macedonians, its reurbanization was distinctly different from the city plans of the Late Antique empire (see Mango, Le développement urbain de Constantinople). And Constantinople was lucky to have even 40,000 people. Most cities were abandoned or reduced to glorified villages (see The Archaeology of Byzantine Anatolia). By 700, for instance, every road leading to the philosophers' city of Miletus had been blocked with marble because nobody needed to take care of them anymore. And so, even in the height of the Macedonian Renaissance, the Persian
Rome had been a monetized empire; yet no coin mint other than the central one in Constantinople survived the seventh and eighth centuries, and no mint would be reopened until Thessaloniki (closed in 629) in 824. In the Byzantine Balkans and Cyprus, the collapse of industry resulted in even the potters' wheel being abandoned for crude hand-shaped pots (called "Slavic Ware").
This seventh-century crisis was transformative not just in the economy, but in politics (the Senate virtually disappeared and the emperors dropped their claims to the Consulate--even if Kaldellis is right about the Byzantine republic, at least the formal trappings of the Republic ended), religion (a general rise in intolerance and enforcement of Orthodoxy), general geopolitical outlook, architecture (there is no Byzantine monumental architecture firmly dated to the seventh century, and even after recovery the architectural tradition of Antiquity was severely impaired), and even understanding of history (the Greek chronicle tradition ends in 627 with the Chronicon Paschale and doesn't start again until Theophanes and Nikephoros in the 810s, and the eighth-century Parastaseis Syntomoi Chronikai shows a shockingly poor understanding of even Justinian's reign, misidentifies the most basic functions of the Antique statuary and monuments of Constantinople, and reduces Septimius and Constantine to figures of legend).
The best analogy might be the Mughals. Legally speaking, the Mughals were Timurids. They identified as Timurids and did not call themselves Mughal. But we don't usually call the Mughals "Timurids of India" because the Central Asian empire of Timur was so different from the Indian empire of Aurangzeb.
Same goes for Byzantium.