These points will be fairly random:
1. It was recognized as the Roman Empire. The Holy Roman Emperor was crowned as such as the Pope, recognized by the Byzantine/ Eastern Roman Emperor as his colleague, etc. By the way, official government documents were printed in Latin down to 1806. I don't get what the "what if" means.
2. The term "Holy Roman Empire" was an invention of Barbarossa for propaganda purposes that for some reason stuck.
3. The term "Roman" really applied to the rulers, not the population. This is confusing. Charlemagne made a bid to be recognized as Byzantine emperor on the grounds that the position was at the time occupied by a woman, in other words vacant. The Pope crowned him (no actual Roman Emperor, East or West, was ever crowned by the Pope). Once the Byzantines replaced a woman with a man, they struck a deal that Charlemagne would be recognized as the colleague of the Roman Emperor in Constantinople, in exchange for evacuating some cities he had occupied that were aligned with them at the time. As Bury pointed out, this wasn't as big a deal for them as it seems to us. After 285, the Roman Empire usually had two Emperors, one in the East and one in the West, and the Byzantines themselves often had more than one person legally holding the title. After Charlemagne died, the title passed to his son Louis and grandson Lothair, and then to whoever had the best claim to be King of (usually anarchic) Italy.
4. After the Treaty of Verdun, Charlemagne's empire was partitioned, and as stated above whoever had the best title to the Italian spin-off was generally recognized as Roman Emperor. After becoming King of Germany, the Saxons conquered Italy and became King of Italy as well. That got them the title. Some, not all, of them chose to make a big deal of it. The Kingdoms of Italy and Germany continued as entities and the Holy Roman Emperor was usually crowned as such, often before he officially assumed the grander title. Other than the Emperor himself, there were no common institutions between the two kingdoms.
5. The Byzantines did call themselves 'Romanoi", and the first Turkish successor state parked on what had been Byzantine territory was called "Rum", but then what was this mixture of Greeks, Armenians, Slaves, and what-not going to call themselves? There was no reason for the people living in the Kingdoms of Germany and Italy not to continue to call themselves "Germans" and "Italians".
If you want people in Western Europe, other than inhabitants of Rome itself, to continue to call themselves "Romans", you need a completely different set-up than the historical Holy Roman Empire. You also need a POD before 1000.
There are three ways to do this, and they are all unlikely. The first is that is that the East Roman Emperor himself sets up a Western colleague and essentially revives the Western Roman Empire, and it somehow becomes more successful than IOTL and expands. The time to do thiw was in the second half of the fifth century, when the idea was considered, but maybe it could be done as late as the reign of Constans Pogonatos. No Arab conquests would help enormously.. The second is that Charlemagne and Louis the Pious both actually want to and are able to keep the empire united, maybe in Constantinople they make Charlemagne the Emperor when Irene is removed, and they import alot of people who can read and write from Constantinople who can staff a bureaucracy. The third is that Otto III, who among Holy Roman Emperors took the title the most seriously, both lives longer and succeeds in reviving the corpse.