To add on to this question:

-Did the existence of the HRE perpetuate or cause a 're-birth' of Roman identity at all in the West? For example in the Kingdom of Italy under the HRE in 1000AD, could your average citizen identify as a Roman?

-In pockets of Italy under Byzantine rule for longer such as the Catapanate, would Roman identify have persisted for longer? Are there any similar anecdotes about rural "Romans" during Italian Unification?

I want to quickly preface this by mentioning that nationalism and a nation as a concept is very very young, starting in the mid-late 1800s, often nationalist movements identified a point in time which they romanticised and emulated. This was often a period that had iconic clothing and are revered, especially in warfare - think Highlanders, Roman Soldiers, Cowboys etc. I can't comment on those children in particular, but I would probably suggest a good question would be whether that identity actually existed for centuries or whether it was born out of a nationalistic movement.

The only example of post-Roman, Roman identity I can think of is in Dumnonia (a nation in south-west England, consisting of modern day Cornwall and Devon). After the Romans left, they probably considered themselves Romano-British, but when the English began attacking at their doorways (invasion was from about the 7th to 9th centuries) they actually became more Romanised in their culture eg. clothing - perhaps it was a romanticised call back to the Roman Era? Similar to how some Hong Kongers romanticises the British Empire after threats from China.

So to expand further, the average Briton at this point in West Country history might have felt relatively Roman, or at least had some call back to the Roman Empire (think ‘the good old days’). They were loosing more and more land to the English invaders and some of them were being persecuted under the Laws of Ine, which were ethnic laws that separated the ‘West Welsh’ ie Dumnonians, from the English, and considered the Dumnonians to be less of a person than the English. They also were on the end of having their language covertly suppressed and the English were attempting to convert them from Celtic Christianity to Catholicism. A lot of Dumnones would have been taken as slaves following the battles with the English.

So not a great time to be Dumnonian - either a commoner or a noble. Perhaps they had some sort of Roman identity because they viewed being in the Roman Empire as better - no threat to their language, religion and there was very little negative Roman impact in Devon and Cornwall. The clothing is the main indicator of this, immediately after the Roman expulsion, the clothing returns to being very Brittonic, but when the English are invading it reverses to having a lot of Roman inspiration.

Bibliography

The Western Kingdom, by John Fletcher

The Archaeology of Late Celtic Britain and Ireland, by Lloyd Lang

Imagined Communities: Reflections on the Origin and Spread of Nationalism, Benedict Anderson

Thought of another side question while I wait for an answer:

-Did Roman identity persist longer in Cyprus than mainland Greece since it 'missed' Greek independence? I know that the Greek Cypriots eventually underwent a similar identity shift with the enosis movement, but I can't find much about the topic since all the scholarly articles are about modern Cypriot identity.

Another question: how did those who identified as Romans assimilate into Greek or other cultures in the 19-20th century?