In particular, were these Romanians claiming to be the successors or to have any kinship to the recently-fallen Byzantine empire?
Interesting OP.
I’m suggesting a double perspective on the self-indentity of medieval Romanians, based on the concept of Romanity.
1.Romanity as both language and origins.
Medieval Romanians were strongly aware that they speak some kind of “Roman” language, that is, the language of the ancient Romans from Rome. There are some three dozens documents of Renaissance travellers in the XVI century reporting that Romanians speak some kind of corrupt Latin which they call Roman and so they were calling themselves, as a nation. Some hundred years later, Romanian Chronicles vehemently represent the same opinion.
Narrowly linked to the linguistical Romanity conscience, medieval Romanians cultivated the Roman origin myth. In the premodern societies origins myths are crucial for the collective identity, every group of people had its own originary myths, the Hellenes, the Romans, the later Germanic Kingdoms, up to the modern nations.
Myths about the own origins and ancestry are essential in every culture, so medieval Romanians had their own identity supporting myth of the origins: they were the remnants of the Roman colonists and legions sent by Rome many centuries ago.
Both the Renaissance travel writers and the Romanian Chronicles report about how medieval Romanians considered themselves being the descendants of Roman colonists and legions.
So, in the Middle Ages Romanians had a strong self conscience of the Romanity of both their language and origins, which they knew to come from that remote ancient city of Rome.
It’s important to underline that this Romanity conscience of the medieval Romanians was not a political one, Romanians didn’t consider themselves as continuators of the Roman Empire, of the Roman civilization or institutions in whatever way. Actually, the first impressions of the Italian Renaissance travellers are quite significant in this sense: they observe that “beside their (corrupt) Latin idiom, these Romanians have nothing Roman left”
2.Eastern Romanity as religious, political, cultural, institutional heritage of The Eastern Roman Empire (Byzance)
From the very beginnings of their states in the XIV century, Romanians reproduce Byzantine structures in the Church, administration, social structure, army, institutions, actually everywhere.
What I call Eastern Romanity, in other words Byzantine civilization, is a very complex institutional structure, political system, religious formation and civilization type encompassing the intellectual hellenistic heritage, the Roman Law and the Orthodox Church.
The Byzantine imprint of the Romanian Principalities was enormous, so that after the Fall of Constantinople, they considered temselves legitimate continuators of the Eastern Roman civilization. To note that the same did Moscow, anyway with much more succes, since they were simply bigger.
A significant term in the international historical research (coined by the Romanian historian N. Iorga) is “Byzance after Byzance”, which seizes this civilizational continuity and irradiation of the Eastern Roman Empire. Iorga demonstrated how much the medieval Romanian States actually were “Byzance after Byzance”.
Other nations in the Orthodox world reclaimed as well being kind of Byzance after Byzance. Actually, as in the case of the (Western) Roman Empire, there was quite a crowdy competition for the symbolical succesion of the Eastern Roman Empire.
Very interestingly, there are no indications that medieval Romanians called themselves Romans as for their appartenace to the Easter Roman civilization (Byzance). Their self-denomination had but linguistical and origins-related reasons. On the other hand, Byzantines called themselves Rhomaioi just to point out an institutional and political continuity with the great Roman Empire, of which they were the only and legitimate heirs.
I think that both Romanians and Byzantines calling themselves Romans is just a series of parallel facts, out of different reasons.
Romanians called themselves Romans not because they were heavily influenced by Constantinople, but just because thy were Latinophones. In other words, while there is no political project behind medieval Romanians calling themselves Romans, Byzantines calling themselves Romans is all but politics.