In the podcast The History of the Byzantine Empire, the narrator said something along the lines that in World War I an American soldier asked a child on a Greek island if they were Greek. The child replied by saying that they were actually Roman.
Understanding that the Byzantine Empire was actually the Roman Empire and that it existed in some form up until 1453, these people that considered themselves to be "Roman" up until the very end surely did not all become Greek, Bulgarian, or Turkish all of a sudden.
In the former Western Roman Empire, did all of the people suddenly identify as Frankish, Gothic, or some other nationality? In the Italian peninsula, the Roman Empire maintained holdings on the peninsula until almost 800. Did these Italian people that had been under Roman control for a thousand years identify as anything other than Roman?
Follow-up question: It seems that the Holy Roman Empire is often mocked by historians. They say that "it was neither holy nor Roman." However, when it was formed in the year 800 with the crowning of Charlemagne, wouldn't there have been many people at that time within the boundaries of the empire that still considered themselves to be Roman? Since Charlemagne was ruling over territory that had been under legitimate Roman control as near in the past as a few decades and, at the most, a couple centuries and contained people that may still consider themselves to be "Roman," then it seems to me that he had a legitimate claim to the title of Roman Emperor.
Self identity is not a simple concept. One could argue that people never stopped considering themselves Roman. You could easily find someone alive today who identifies as a Roman.
One problem we have is definitions. What is a Roman? Simply a resident of the city of Rome? Or do you have other qualifiers, such as the Latin language or maybe ancient Roman citizenship? If we are considering self-identification you can see that the definition will be very broad.
Remember that these identities are fluid and always have been. You can find someone in Cincinnati today who might identify themselves as a Buckeye, an Irish-American, an African-American and an American – all at the same time, or perhaps different identities in different contexts. Here in Finland you can find someone who calls herself a Finn, a Swedish-speaking Finn, an Ålander, a European - again, all in one person.
The same was true throughout Roman history. There were people who considered themselves Romans as well as Goths, for example, or Romans as well as Greeks. There were also local identities which people identified with, such as cities or regions or even local warlords. This became more relevant during the fall of the West and as power devolved from imperial to local sources.
There have been finds of ‘barbarian’ chieftains in North Africa using Roman identifiers, like Roman titles, the Latin language, and various ‘Roman’ attributes such as civic duty, urban life and a stoic demeanour. However, these ‘barbarians’ had no political ties to any authorities in the Roman Empire. Meanwhile you have people undoubtedly Roman, like soldiers in the Army in Late Antiquity, rejecting some of these ‘Roman’ attributes and claiming ‘barbarian’ ones like animal savagery and hot-headedness.
Throughout all of this the concept of ‘Romanness’ never disappeared. It evolved and changed and was a fluid concept. It was useful in some contexts and worthless in others.
One interesting source for this in relation to the end of the West is Guy Halsall’s Barbarian Migrations and the Roman West 376-568.
if i remember correctly, Robin's (from the history of byzantium) story was not about american soldiers in WWI, but greek soldiers in the XIX century.
You're first mistake was believing that Byzantium was objectively roman.