His family name was Julius (because he was from the family of the Iulii). Caesar was a title and since Augustus became part of the name when the position was taken up.
Caesar was a title
Caesar became a title after the collapse of the Republic, largely because the Augustine line carried it forward as proof of legitimacy. There was no position of “Caeser” within the Roman Republic, just a family name dating back to Julius Caeser’s great-grandfather.