Actually this is history’s greatest example of what you might call the “euphemism treadmill.”
Roman tradition was extremely hostile to the concept of monarchy; the founding myths of the Roman Republic were oriented around the wickedness of kings (particularly the last king, Tarquin the Arrogant). Distrust of monarchy was the reason that the Romans didn’t have a single executive officer for any level of government — all the magistracies, right up to the conuslate, were plural. The greatest heroes of Roman history were men who were asked to temporarily assume absolute power in an emergency and then who meekly resigned it and went back to being regular citizens when the crisis passed.
Even Julius Caesar, when he was in all but name sole ruler of the Roman world and his reputation with the common people was quasi-divine, was hissed out of the forum when his friends offered him a crown. As Shakespeare has Marc Antony say
You all saw that on the Lupercal feast day I offered him a king’s crown three times, and he refused it three times. Was this ambition?
Actually it was ambition— most historians believe Caesar asked his friends to stage the crown incident as a trial balloon, and he was extremely irritated that it turned out to be a PR disaster.
Against that background Caesar’s adoptive son and heir Octavian — better known to history as Augustus, the first emperor — had a very interesting political problem to solve. Caesar’s rise had kicked off a generation of brutal civil wars, which only ended when Octavian defeated Marc Antony and Cleopatra and became the undisputed ruler of the empire. He was, however, leading a Roman state that had been decimated by a generation of internal warfare. To consolidate his own power he needed the support of the traditional elite — the aristocrats who were bitterly opposed to monarchy in theory, and most of whom had lost family members (often, to him) in the long civil wars.
Augustus wisely chose to hide his absolute power behind a demure facade of continuity with the Roman past. Instead of adopting the trappings of monarchy he pretended to be just another senator; the Senate continued to meet and debate, new consuls were elected, and the social forms of the old Republic were followed scrupulously… at least, in public. Nobody was really fooled, but this symbolic concession allowed breathing space for Rome to adapt to the new realities.
The title “Caesar” was an essential part of the game. Old Roman manners disdained the fawning, titles, and ceremonies of Greek or Asian monarchies. In the “good old days,” that Augustus was trying to simulate, any Roman citizen could approach the most powerful man in the state and address him by name — there was no Roman equivalent of ‘your highness’ or even ‘sir’; for a self-respecting Roman it was merely “good day, Gaius Julius. “ So, one called Augustus “Caesar” in order to play a part in the pantomime of old Roman manners that was the essence of early imperial politics.* This was as obligatory— and as fundamentally dishonest — as the epithet “Comrade Stalin.” Augustus accumulated many accolades — including the name “Augustus,” roughly “the important one” —but he never held a permanent, well defined office. Thus, his adoptive personal name became the most important and common way in which he was addressed or discussed.
The first five emperors were all members of Augustus’s family. To greater or lesser degrees they carried on the fiction that he had established, so for the first century of the empire there was always a Caesar in charge at Rome even though none of these men was the biological son of his predecessor. The personal name, however, was quickly morphing into a title — indeed, outside of Italy it was more like a magic incantation . In the provinces, where old Roman ideas mattered little, the emperors expected their subjects to offer them worship in specially dedicated temples. Theoretically this was an offering to the emperor’s genius or guardian spirit, but it’s obvious that the distinction was academic to the huge mass of non-Latin speakers who only knew that they offered prayers and swore oaths to “Caesar”. It believed that even in the first century Germans were using Kaiser to refer to the far-off ruler of the Romans.
With all of that in the air, even if a new emperor had wanted another title there huge momentum behind the one-time euphemism. When the throne finally passed out of the Julio-Claudian family (in 69, the Year of the Four Emperors) the victorious claimant executed a senator who insisted on addressing him by his given name rather than the honorific “Caesar”. When the Persian emperor wrote to him addressing him by his birth name, he wrote back omitting all of the Shah’s titles. The transition from name to title was complete, a bit less than a hundred years from Augustus’s day.
New titles came around in time — in the second century it became acceptable to call an emperor a basileus, a “king” in Greek or a dominus, a “lord” in Latin. Increasingly baroque locutions started to proliferate after the 5th century but “Caesar” remained an important title (though not always the highest in precedence) down to the end of the Byzantine empire in 1453.
After such a long and successful run the word was inevitably picked up in a score of other languages. On the even of world war one you had a Kaiser in Germany and another in Austria; Tsars in Bulgaria and Russia; and even the Sultan in Istanbul was the Kayser-i Rûm. The last of the Caesars finally laid down his title in 1948, when George VI of England, the Qaiser e Hind surrendered his imperial title on Indian independence.
See also
Why are Roman rulers from Augustus (sometimes even Julius Caesar) called emperors by modern people?
The Wiki page has a pretty good overview of the later evolution of imperial titles.
*One thing you definitely did not do was to address Augustus by his pre-adoption name, Gaius Octavius. A lot of Octavian’s political cachet stemmed from his association with Julius Caesar and he insisted on always and only being “Gaius Julius Caesar” after his adoption, even though Roman tradition would have had him become “Gaius Julius Caesar Octavianus”