Wouldn't Ceasar be much closer to his last name,though? It's definitely not his personal nickname.
His father was 'Gaius Julius Caesar' his grandfather (supposedly the first 'Ceasar') was 'Sextus Julius Caesar'.
The Cognomen is presumable something that evolved as a way to tell apart people belonging to different branches of the same family/clan because the number of first names was very limited.
Nicer than his other nickname 'adulescentulus carnifex ("teenage butcher") ' I guess..
But yeah, the cognomen seems weird/hard to understand e.g. Pompey's dad was called Strabo but he seems to have never used that name himself, while other like Ceasa, Metellus or Cicero were pretty much just surnames.
Romans probably found their naming system as confusing as we do so they started using nicknames ad hoc.
It was even more awful if you were a woman, you technically didn't even have your own name until the imperial period. You and your sisters would just be called Aemilia I, Aemilia II, III etc (or the 'elder'/'younger'). if your father was belonged to the Aemilius clan for example...
In inscriptions (short): CAIVS IVLIVS CAESAR
In speaking he would (in English;)) rather formally introduce himself as "Gaius of the Iulii, called Caesar".
His mother called him Gaius, his peers called him Caesar once he got or choose?) this name.