I can't speak for the Carolingian empire or high medieval Holy Roman Empire, but the Byzantines certainly had a problem with the Ottonian HRE calling itself Roman. Luitprand of Cremona, an ottonian diplomat, records these interactions between himself and the Emperor Nikephoros Phokas (note that he was treated quite badly by Nikephoros, and so the account is probably quite biased and exaggerated):

" He called you not emperor, which is Basileus in his tongue, but insultingly Rex, which is king in ours. I told him that the thing meant was the same though the word was different, and he then said that I had come not to make peace but to stir up strife."

"I wanted to answer and make such a speech in our defence as his boasting deserved; but be would not let me and added this final insult: "You are not Romans but Lombards.""

"But, to increase my calamities, on the day of the Assumption of the Virgin Mary the holy mother of God (August 15), there came-an evil augury for me-envoys of the apostolic and universal pope John, through whom he asked Nicephorus ,the emperor of the Greeks " to close an alliance and firm friendship with his beloved and spiritual son Otto "august emperor of the Romans." Before the question as to why- this word, this manner of address, sinful and bold in the eyes of the Greeks, did not cost its bearer his life-why he was not annihilated before it was read, I, who, in other respects, have often shown myself enough of a preacher and with words enough at my command, seem dumb as a fish! The Greeks inveighed against the sea, cursed the waves, and wondered exceedingly how they had been able to transport such an iniquity end why the yawning deep had not swallowed up the ship. " Was it not unpardonable," they said, "to have called the universal emperor of the Romans, the august, great, only Nicephorus: "of the Greeks"'; and a barbarian, a pauper: of the Romans'? Oh sky! Oh earth! Oh sea! But what," they said, " shall we do to those scoundrels, those criminals?" They are paupers, and if we kill them we pollute our hands with vile blood; they are ragged, they are slaves, they are peasants; if we beat them we disgrace not them, but ourselves; for they are not worthy of the gilded Roman flail and of such punishments. Oh would that one were a bishop, another a margrave! For sewing them in sacks, after stinging blows with whips, after plucking out their beards or their hair, they would be thrown into the sea. But these," they said, " may continue to live; and, until the holy emperor of the Romans, Nicephorus, learns of this atrocity, they may languish in narrow confinement."

"Hear then! The stupid silly pope does not know that the holy Constantine transferred hither the imperial scepter, the senate, and all the Roman knighthood, and left in Rome nothing but vile minions, fishers, namely, peddlers, bird catchers, bastards, plebeians, slaves. He would never have written this unless at the suggestion of your king; how dangerous this will be to both-the immediate future, unless they come to their senses, will show.""

Keep in mind that Nikephoros was famously hot headed and undiplomatic, and evidently so were the people he appointed to manage diplomacy. The two empires also had bad relations because of contested duchies in southern Italy, which didn't help. Normally, the Byzantines had a lot more diplomatic tact, and the viewpoints expressed in Luitprand's report would be suppressed.