When talking about the collapse of the Western Roman Empire, it's important to remember that although the Western portion of the empire collapsed in 476 AD the Eastern Roman Empire(also known as the Byzantine Empire) continued to exist until 1453.

Also although the Western and Eastern Roman Empires are currently referred to by separate names, and had separate bureaucracies armies etc. ,they were still considered to be the same country and both the Western and Eastern Emperor were considered to be Emperors of the whole Empire. So even after there was no longer a Western Emperor, there continued to be an Eastern Emperor who called himself and was widely recognised as the Roman Emperor. Therefore contemporaries would not have considered the fall of Rome in 476 AD to be the end of the Roman Empire.

Another reason why the fall of Rome in 476 isn't viewed as significantly by contemporaries is because the last Emperor Romulus Augustus was viewed as being a usurper by most of the Roman Empire. Most of the Roman Empire recognised Julius Nepos as the Western Roman Empire. In fact Odoacer who conquered Rome in 476 AD viewed Julius Nepos as being the rightful Emperor and accepted Julius Nepos as his sovereign. So even after the fall of Rome, the rulers of Rome and Italy still considered it to be a part of the Roman Empire.

This also ties into why most of Western Europe still considered itself to be a part of the Roman Empire, the way in which the Roman Empire lost control of Western Europe was through foderati agreements. The Roman Empire during the 4th and 5th centuries would sign agreements with barbarians that would allow the barbarians to settle on land in the Roman Empire, however the barbarians would have to swear loyalty to Rome, and originally would have had to send soldiers to support the Roman military. Although these areas were nominally still a part of the Roman Empire the Emperors had no direct control over these areas.

During the 5th century Roman Emperors were forced to sign foederati agreements in order to make peace with their enemies. The Visigoths who had ruled over most of the Iberian peninsula had a foederati agreement with the Romans and although they ruled over Iberia since the late 4th century they didn't declare independence until 475 AD. The Franks who ruled over most of France also used to be foederati of the Romans. The Ostrogoths who ruled over Italy from 493-540 were also foederati.

In addition to the foederati there were also other regions that considered themselves to still be a part of the Roman Empire, but could not be reached by imperial forces and therefore effectively became independent. These were the Mauro Roman Kingdom, which ruled over parts of modern day Morocco and Algeria. The Mauro Roman Kingdom typically allied itself with the Byzantine Empire, and some of their rulers like Stotzas were originally from the Eastern Roman Empire. However towards the end of the Kingdom, their last King Garmul tried to invade Byzantine controlled North Africa, his death in 578 AD led to the Kingdom being reabsorbed into the Roman Empire.

Another region that still considered itself to be a part of Rome was the Kingdom of the Romans which ruled over Northern France from 457-486. Their first ruler Aegidius was the magister militum of Gaul. And although they are referred to as a Kingdom, they considered themselves to be a part of the Roman Empire with their rulers taking the title of Dux. They operated independently because southern France had been conquered by the franks and they couldn't receive or give assistance to the rest of the Roman Empire. The Kingdom of the Romans was eventually conquered by the Franks in 486 AD.

Finally the Roman Empire had abandoned Britain in 410 AD, this led to most of Britain with the exception of Wales and Cornwall being conquered by Germanic tribes during the following centuries. Due to a lack of written sources its difficult to determine how the Britons viewed themselves, but there is some evidence that the early welsh kingdoms might have viewed themselves as still being a part of the roman empire.

In summary, there were still regions that still considered themselves to be a part of the roman empire. These were the provinces of the Eastern roman Empire, Kingdoms that had foederati agreements with the Roman Empire, and areas on the periphery of the Western Roman Empire that were cut off from the rest of the Roman Empire.

Yes, indeed there were! The latest example I know of, just to start, is the province of Africa. Up until the conquest by Islamic forces in the 700s, the people of North Africa were still calling themselves Roman. They were semi-independent and enjoyed Exarch status with Constantinople. You can read more about the last leaders here.

All the same, it's not hard to argue the Franks were effectively doing this. In Gaul, the Frankish rulership were descended from Frankish Foederati who had served in the Roman Military. The conquest of Soissons wasn't so much a conquest as a coup. After the Franks cemented their rule in Gaul, they actually were given a status akin to Exarch, though never outright stated as far as I know. It just had the same exact political structure as the Exarchs. Constantinople would send over to the Franks various governors, advisors, and occasionally generals to help the Franks rule in a Roman style. This was part of a kind of compromise between the Franks and Romans. Franks got to live on the land and be kings, while Romans got to elect the kings and guide the king with a "mayor of the palace", or sometimes called a Praetor instead of Mayor. These two positions of Rex and Praetor/Mayor were elected positions. The population of Gaul would have a general assembly and elect the leader they felt was best for them. The kings generally just sat on their buts and acted as a symbolic monarchy, while the Mayors did all the work. This is attested to in the work of Einhard in his "Vita Karoli Magni", the life of King Charles. Einhard documents the electoral campaign of Charles and his brother running for the position of Mayor and King. They win the election but his brother decides to quit the position soon after taking office, and Charles becomes elected to both offices.

Between the use of Roman titles, roman languages, officials from Constantinople, and the paying of trade taxes to her, it's safe to say the Franks considered themselves Roman. 400 years later during the 4th crusade, when the French and Venetians conquered the Eastern Roman Empire, the charter they wrote defined them as "Latin people" who were "Partitio terrarum imperii Romaniae", or partitioning the Roman Empire. Baldwin I of this Latin Empire was a Frenchmen, and didn't see anything wrong calling himself a Latin Roman Emperor.

Between the fall of Western Europe, to the crowning of King Charles into Emperor Charles, the Papacy was also ruled by the Byzantine authorities. The Emperors selected the Pope, who in turn selected the kings of western Europe.

It is for this reason that it's not right to say the Roman Empire "collapsed". The Dark ages mark a period of transition starting from Romans giving high authority to foreign born German immigrants, until the crowning of Charlemagne firmly establishes the sovereignty of the west apart from Constatninople's rule.

It's much more accurate to say the Roman Empire simply evolved into a shared ownership of the west between greeks and germans, with the germans becoming latinized, and the Greeks loosing interest.

An interesting story recounted by Peter Charanis (a famous Byzantinist who was born in the island of Lemnos in 1908) highlights this:

When the island [Lemnos] was occupied by the Greek navy [in 1912], Greek soldiers were sent to the villages and stationed themselves in the public squares. Some of us children ran to see what these Greek soldiers, these Hellenes looked like. ‘‘What are you looking at?’’ one of them asked. ‘‘At Hellenes,’’ we replied. ‘‘Are you not Hellenes yourselves?’’ he retorted. ‘‘No, we are Romans."

(Taken from Hellenism in Byzantium by Anthony Kaldellis).

I don't think they believed themselves to be a part of the literal empire, but certainly there were still people for a long time who considered themselves to be Roman people.

John Romer in his masterclass work “Byzantium: The Lost Empire” has a great part where he talks about the “fall” of Rome was really an concept created later on by the Christian Popes of a pagan Rome falling and a Christian Rome rising. What really happened was when Constantine made Constantinople the new Roman capital it took all of Rome’s wealth with it. Rome was modeled after the great cities of the Mediterranean and Romer explains that “Rome didn’t fall, it simply went back home again.” I would imagine that although Rome was sacked it’s empire still existed in Byzantium and still collected taxes in many provinces.