Because it's a Western bias that started after the German historian Hieronymus Wolf published in 1557 his work Corpus Historiæ Byzantinæ, and it caught on.

The name the Byzantines had for their nation was Roman Empire, Empire of the Romans or Romania, depending on the situation. A quick look at Anna Komnena's Alexiad will give you hundreds of examples. However, in the West the term "Roman" started being associated to the idea of the Carolingian Empire of Charlemagne (crowned in 800 by the Pope, taking the title "most serene Augustus crowned by God, the great, peaceful emperor ruling the Roman empire"). The idea was later revived by Otto the Great, the founder of the Holy Roman Empire as we know it. The Popes even made a forgery, the Donation of Constantine, to support the Pope's authority in this matter.

Let it be said that before Charlemagne's coronation the relationship between the Pope and the Byzantines wasn't great, and the Pope saw the Carolingians as safer and more reliable protectors.

From that moment onward, in Western Europe the title "Empire of the Romans" was officially reserved for the German empire. As BoraFan has mentioned, the term "Empire of the Greeks" started being used for the Byzantines.

And no, that didn't sit well at all in Romania.

To give a first-hand example, Liutprand of Cremona was an ambassador from Emperor Otto to Constantinople and he had a discussion about the nature of being 'Roman' with the Byzantine Emperor himself. You can read more about his diplomatic mission in 968 in this link, but here's the part about the Byzantines' reaction when they received a letter by the Pope (that was intended to be a gesture of goodwill) in which they were called Greeks instead of Romans:

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"';-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."

So, yes, being Romans was serious business for the Byzantines.

If Western historians started using the name 'Byzantine' was probably due to custom (helped by the fact the Byzantine Empire didn't exist anymore to complain) and an increasingly bad reputation of the Byzantines among Western writers, who nevertheless admired classical Rome.

Ironically, their real name was more respected by their Muslim enemies. Their name for the Byzantines, Rum, comes precisely from the word "Roman". There is even a surah in the Quran called "The Romans" (Ar-Rum), in which it is declared that despite their initial defeats, the Romans would win the war against Persia (it's about the Byzantine-Sasanian War of 602-628). When the Turks invaded Anatolia, they founded the "Sultanate of Rum" and later, after conquering Constantinople, the Ottomans used, among many other titles, the name Kayser-i Rum.

Whilst the Byzantines referred to themselves by various names referring to their Roman origins, much of Europe began to associate them over time with the Greek ethnicity of many of their inhabitants after the fall of the Western empire. To outsiders the Byzantine Empire was commonly known as "The Empire of the Greeks", and several centuries after their dissolution in the 1400s the term "Byzantium", referring to the original name of the Greek colony which became Constantinople, was popularised by historians and thereafter gained usage as distinctly referring to the ERE.