This is indeed true, since the Roman Empire did not collapse in 476. Parts of the empire fell beyond centralised control or were seized by warlords, but these new rulers generally tried their hardest to maintain the Roman form of government, since you know, it collected taxes, provided security and maintained a justice system - why would any king want to get rid of that? Though the empire itself was gone, the system, or at least parts of it, that the empire embodied remained for many people. The same was of course even more true in the eastern half of the empire, since the emperors ruled Constantinople continuously until 1204 and with a minor interruption until 1453. Admittedly the Arab Conquests led to the loss of many wealthy provinces, such as Egypt, Syria and Palestine, but the conquerors again didn't rip the system apart, since in most cases the existing officials were kept in place, people paid taxes to the same people and used the same coins. For all intents and purposes, Roman life continued to flourish under Arab rule at least until the end of the seventh century. For the empire itself, though it was very different in 1453 compared to the empire of 476, it was still the same Roman Empire, as there was no reason for the inhabitants to think otherwise. It might look different on the surface, but change is inevitable and the Roman world had evolved constantly, as we only have to compare the empire of 476 with the Roman Kingdom of the eighth century BC to see how dramatic this process can be.

The Eastern Roman Empire also continued to have a say in western affairs. The man who overthrew Romulus Augustulus in 476, Odoacer, for example nominally gave his allegiance to Emperor Zeno in the east and pretended to rule Italy in the emperor's name. He was later overthrown by Theodoric of the Ostrogoths, whose formative years were spent in Constantinople and who was even a consul, and he may even have been sent by the emperor to reclaim Italy for the empire. His kingdom was an exceedingly successful one, to the extent that there are good reasons to think that he was trying to restore the Western Roman Empire. The Senate had a say in government, consuls were still being nominated and Theodoric was compared to great emperors such as Trajan and Valentinian. Other contemporary rulers, such as Clovis of the Franks and Gundobad of the Burgundians, also sought legitimacy by acquiring new titles from Constantinople; remember that they were basically ruling Romans in their new kingdoms, so placating the population was essential - what better way to do that than to rule as Romans would? Both individuals were also involved in Roman government before it fell in the west, as Clovis had essentially inherited the military governorship over Belgica, whilst Gundobad was one of the kingmakers who played the game of thrones in the last decades of the Western Roman Empire.

Constantinople's hold on the west was strengthened further after Theodoric's death, as the ensuing collapse of the Ostrogothic world order (Theodoric had basically acquired informal authority over the Vandals in North Africa and the Visigoths in Spain) provided an opportunity for Emperor Justinian to strike back. Vandal North Africa fell quickly and direct Roman rule would continue until the end of the seventh century. Italy took longer, but total victory was achieved by 554 and although peace was quickly shattered by the Lombard invasion, parts of the peninsula belonged to the empire until the eleventh century. Along the way, a slice of Spain was also conquered amidst a Visigothic civil war, which was only retaken by the Visigoths in the 620s. Taken together, the majority of the Mediterranean world (between southern Spain all the way, through North Africa, to the Italian coast where it meets Francia) was Roman until the empire was faced with a new crisis in the seventh century. Even after that, a good chunk of the Balkans, the majority of Anatolia, various enclaves in Italy and numerous Mediterranean islands remained under direct Roman control. Even when these regions were lost, Constantinople itself was still the seat of the emperor until 1453, so yes, it is true that many people still lived within the empires for centuries and in some cases nearly a millenium after 476.

The big thing to keep in mind is that the traditional date for the fall of the Roman Empire in 476 CE is more appropriately considered the fall of the Western Roman Empire. The Roman Empire had been split between West and East prior to the Fall of the Western Roman Empire, though this was not a split into multiple states as much as a practical solution to the difficulties of administering a state as large as the roman empire from a single location, and the Emperors in West and East were generally of the same dynasty.

To help cement this ideal, Constantinople was even declared the Second Rome when this split was first set up, and was given institutions to match those of Rome (while Italia was deprived of some of its traditional privileges (primarily tax-exemptions) to further equalize Rome and Constantinople without affecting the budget too negatively.)

After the Pope crowned Charlemagne Roman Emperor (the first time a Germanic King was crowned Roman Emperor by the Pope, but hardly the last) the legitimacy of the surviving eastern half of the Roman Empire as a valid successor state of the Roman Emperor became increasingly questioned by the Western Christians.

This especially became the case after the Western and Eastern Catholic churches split into the Roman Catholic and Eastern Orthodox churches in the Great Schism. Two Christianities each with their own Roman Empire.

The people in the Eastern Roman Empire never stopped considering themselves Roman. There was an undisturbed continuation of Roman policies and Roman-style governments in the East after the city of Rome fell, and even after Ottoman Conquest of Constantinople the Ottoman bureaucrats still considered these people Romans. This did spell the end of the Eastern Roman Empire, but this was in 1453 CE, very close to a full millennium after the fall of the Western Roman Empire.

Because Western Christians did not consider the Eastern Roman Empire a legitimate successor to Rome, it is frequently not treated as such by traditional narratives of the Fall of Rome, and many texts (though no contemporary ones) refer to it as the Byzantine Empire. In practice, the people of that Empire tended to refer to it as the Roman Empire (and thus themselves as Romans) while the people in the West tended to insist on referring to it as a Greek rather than Roman polity. (Not entirely without basis, the Eastern Romans spoke Greek and had massive Greek cultural influences.)

So in short, if by 'the Fall of the Roman Empire' you refer to the fall of the Western Roman Empire then yes, for nearly a thousand years people still believed to be in the Roman Empire, but there were good reasons to believe so.

A seminal source to read on your question is 'Mohammed and Charlemagne' by Henri Pirenne. In it he charts the various ways in which Roman culture both endured and changed during the centuries following the traditional "collapse" vis a vis economics, politics, language, and other trends.

Subsequent archeological evidence has challenged some points of his thesis but the work is still remarkably insightful and illuminating.