I wonder how the common people of the west would think about that. They've never identified themselves as Romans so the politicians or rather rulers of the west would only moot that point.Finally, having said all this, this setup could open the door for the various western Europeans to claim Roman identity. “We’re culturally French/German/whatever, but politically Roman.” But the counterargument to that would be “Roman is a political identity, and the continuity of that polity was in Constantinople.”
That said, now that you say it, I remember reading one moment in another TTL like this, with one rumite sultan claiming he was Roman, and the response of the Trebizodian Roman emperor was insulting the sultan for such b***********. Would the Romans of this TTL be the same?