What did the title functionally achieve for Charlemagne? I doubt it meant much to the Saxons or Frisians, and it didn't make him an equal to the Eastern court.
In a way I think it was just a formalization of his role in keeping Rome free of the Lombards.
Charlemagne was a bit interested in all the Roman stuff but rather seen himself as the successor as King David than the successor of the emperor Constantine. The rebirth of the Roman Empire was way more a willingness and a dream of the educated court of Charlemagne (especially inside the Palatine Academy) which advised and interested him in consequence.
Frankish and Saxon subjects of Charlemagne were not much interested into it. And in Charlemagne's mind, all this story did not stayed much in regard of his full rule.
The Franks viewed The Roman Empire as that empire in the East which call us barbarian and claim suzerainty over us.
In fact, that's wasn't true:
Two major events contributed to universalize the understanding of what is a Roman, allowing (educated) Franks to perceive themselves as being Romans, not as inhabitants of Rome but as Christians:
1. Under Hadrian's rule by numerous and long travels of the emperor, we assist to a shift of the perception of the Empire from a Roman then Italian hegemony (Rome was no longer the capital since a long time) to an unified and œcumenical Empire [1]. 2. This prepare the roots for Caracalla' edict, of 212. Since this edict, all free men of the Empire were granted Roman Citizenship, so the old divide between Roman citizens and Empire's simple subjects did disappears and Roman was now longer perceived defining all inhabitants of the Empire (a new divide did appears between Honest men [honestiores] and Humble men [humiliores]) [2]. 3. In a way, Roman Empire considered that it was the sole State in the world, and Roman Emperors did not have any Foreign Affairs ministers...[3] 4. With the help of conversion to Christianity, initiated by Constantine since 313, and strengthened by Theodosius and Gratian who instituted the Christian Church as State Church, the Empire became the Christian Empire, something which gave it its universal aspect with the responsibility to preach Gospels to the whole world and to prepare mankind to the rule of God [4].
Hence, at the Time of Charlemagne, Roman and Christian were perceived meaning the same thing. In the West the feeling to have been part of the Empire was still present but very few were aware that the Empire was defined being the political entity of a Greco-Roman cultural community ; it was rather and mainly perceived being the political entity of a Christian religious community: a community uniting the African Augustine of Hippo, the Italian Boethius, the Hispanic Isidore of Seville, the Anglo-Saxon Bede... [4]. Interestingly, in the East, Roman and Christian similarly was perceived being the same concept, but under the word for Greek. There, the emperor was presented being like a thirteen apostle, legislating for the Church in the East [5].
In addition to those two events, two others contributed to make the Eastern Roman Empire being perceived as being rather the kingdom of Greeks (rather as "The Roman Empire"):
If there was, between 395 and 480, two emperors: one in the western part of the Empire (pars occidentalis) and another from the eastern part of the Empire (pars orientalis). This timespan end with the murder of the last emperor in the western part of the Empire, Flavius Julius Nepos, by his soldiers. After him, Flavius Zeno, previously emperor in the eastern part of the Empire, become the sole emperor. But 1. In 629, Flavius Heraclius take the title of king (basileus) rather than of emperor (augustos) and 2. in 797 Irene Sarantapechaina, a woman, become "king" of the Greeks. Basileus now became synonymous to augustos to Greeks but not necessary to others peoples of the former western part.
The terrible way Irene came to power, the fact that she was a woman, the disastrous violent controversies over icons, the bad opinion toward Greek among peoples of the former western part of the Empire (especially in Italy), the "king" meaning of basileus (As I've said, Byzantium was then called the "kingdom of the Greeks" in the west), the political situation in Rome and the prestige of Charles, sole king of the Franks, allowed this last one to become in situation to be accepted as the new emperor in a western part of the former Empire.
And not during Charlemagne but later, during mediæval Holy Roman emperors, the link between the Empire and the Roman Empire will be legitimized by the combination of the
translatio imperii concept with the prophecy of Daniel:
1. the translatio imperii considerate that there is a transition of the imperial authority from east to west. After the Babylonian (First) Empire, there was a Persian (Second) Empire, then a Macedonian (Third) Empire and finally a Roman (Fourth) Empire. 2. However, according to the prophecy of Daniel, there will be only Four empires before the apocalypse so as long as the Roman Empire exist, the end of times is not yet to arrive... As the world hasn't ended so the Roman Empire hasn't fallen. Therefore, someone still possesses and exercises the Roman Empire's authority.
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Returning to Charlemagne: the
basileus Irene of Athens was very soon overthrew by her grand treasurer, Nicephoros, who was proclaimed
basileus by the army. The West (including the Pope & the Emperor in the West) then ended to see the imperial throne is the East being vacant. As emperor, Charlemagne initially and at the end never tried to show himself as a superior to the
basileus, avoided to insist too much on the Roman aspect of his own titles and called the Byzantine emperor "my brother".
To the eyes of Charlemagne and of his entourage, there was two parallels empires : a western Empire and an eastern Empire (while in the ancient world there was one Empire and two Emperors).
In return, the Greek were reluctant to recognize Romanship of the emperor in the West: Michael Rhangabe recognized that there was an empire in the West but did not recognized it being the rebirth of the Roman empire in the West.
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[1] V
EYN (P.),
L'Empire gréco-romain, Paris, 2005, I., 13, citing, about the exceptional travels of Hadrian's: H
ALFMANN (H.),
Itinera principum : Geschichte und Typologie der Kaiserreisen im rômischen Reich, Stuttgart, 1986
[2] V
EYN (P.),
opus citatium, I., 12. and F
AVIER (J.),
Charlemagne, Paris, 2013, XVIII, Le Royaume et l'Empire.
[3] V
EYN (P.),
op. cit., I. 13, citing M
OMMSEN (Th.),
Staatsrecht, III., 1, p. 826 and M
OMMSEN (Th.),
Histoire romaine, II., p. 751, 769 and 781.
[4] F
AVIER (J.),
op. cit.
[5] F
AVIER (J.),
op. cit, Paris and "Byzantine Greeks",
Terminology, in
Wikipedia, citing E
ARL (D. C.).
The Age of Augustus. New York, 1968: Exeter Books (Paul Elek Productions Incorporated), p. 148. and quoting H
ARRISON (Th.),
Greeks and Barbarians. New York: Routledge, 2002, p. 268: "Roman, Greek (if not used in its sense of 'pagan') and Christian became synonymous terms, counterposed to 'foreigner', 'barbarian', 'infidel'. The citizens of the Empire, now predominantly of Greek ethnicity and language, were often called simply ό χριστώνυμος λαός ['the people who bear Christ's name'].".
[4] F
AVIER (J.),
op. cit.
