The Byzantine Empire = Rome? What's the consesus among historians?

That statement is completely correct, you can quibble a bit with the terminology of "Eastern Roman Empire" but the so-called Byzantines were unambiguously a part of the Roman empire that did not fully fall until 1453 (or a bit later if you include Trabzond, Mystra, etc). To give a somewhat less ideologically fraught example, in 1127 the Jurchen captured the city of Kaifeng, capital of the Song Dynasty, along with the Emperor and members of his court (and much of northern China). The remainder of the court regrouped at Hongzhou, where they flourished for the next hundred odd years. It would be pretty odd to say that the court at Hongzhou was "not really Chinese" or "not really Song", and that is pretty comparable to the Byzantine/Roman situation.

Now to be fair, there is a bit of a wrinkle, in that the reason we still care about the Romans is that they managed to unify the entire Mediterranean basin and a decent chunk of Europe for four hundred years under a single political unit, something that never happened before and would never happen after. After the collapse of the western half of the empire that is gone, and the closest it came to coming back would be under Justinian, whose wars arguably put "Late Antiquity" in the grave, and the Umayyad Caliphate, which ran out of the steam pretty quickly. But that doesn't mean the Byzantines were not "really Romans", they were, just with a rather less impressive empire.

I think the issue is that "Roman" is still seen as something aspirational rather than something factual. To skirt somewhat around the twenty year rule, one American political commentator described a particular politician as "a noble Roman out of Plutarch, which is to say, a man of far-visioned republican virtue who was also, in the Roman style, shockingly barbarous; and, in this way, he was the most American American who ever lived." On the face of it this is just kind of goofy, but it really speaks to a somewhat strange mind virus in western culture that the Romans were somehow imbued with particular attributes that were exemplary of some particular thing. If you drop that than the question of whether x was "really Roman" kind of loses its punch.