Given that the Byzantine empire permanently lost the city of Rome in the 800s it must have felt strange calling yourself a "roman" when you don't even own your namesake city.
How did the Byzantine people feel about that odd naming...thing.
How did foreigners see This?
Did Byzantine citizens really had that Roman idendity in all provinces?
While many of the educated no doubt lamented over the loss of Old Rome, as the Byzantines called it, it caused no identity crisis.
Just because the Romans originated in Rome, doesn't mean that they stopped being Romans when they didn't have it anymore. By this period Being a Roman had nothing to do with geography, instead being defined by being a Roman citizen, being a christian, and practicing "roman customs". The Romans gave their name to the land they ruled, not the other way around. Wherever the Romans lived and was Romania.
That being said, it doesn't hurt that Constantine invested so much energy into making New Rome (aka Constantinople, but referred to as either in the middle Byzantine period) look like the natural successor to, and even continuation of, Old Rome, which additionally justified their Roman identity. Old Rome had a senate, new Rome got a senate. Old Rome had 7 hills, New Rome had 7 hills. Old Rome had a Latin speaking aristocracy, New Rome had much of that aristocracy transferred to it (they stopped speaking Latin eventually, but the Byzantines knew full well the boost in legitimacy that having an aristocracy descended from that of Old Rome granted). Old Rome had a circus, New Rome got a circus. Etc.
There was also rhetoric prominent in late antiquity that the city of Rome had expanded its borders to the entire empire in a metaphorical sense (IE it unified the culture, identity, and practices of its citizens, undoing the old order of the independent and proud Polis), which could theoretically be used to argue that the Byzantines never really lost the rhetorical "city" of Rome. I haven't see this argument used in a primary source from the middle period though.
Source: Anthony Kaldellis: "Hellenism in Byzantium"
Primary sources that best illustrate the confidence of Roman identity in middle Byzantium:
The History of Leo the Deacon, on the reigns of Nicephorus Phocas and John Tzimiskes
The "Strategikon" of Kekaumenos, more specifically "Oration of Admonition to an Emperor"
The History of Micheal Attaleiates, on the emperors of the 11th century up to the late 1070's