The same way Gaius Julius received it as an inheritance from his forebears. OP is totally right that the name and its power emanates from Octavian’s prestige and domination of the Roman world for near half a century.
Yeah, but it didn't become a title until the Julio-Claudians started using it to designate their heirs. When Augustus took it on, it was just a powerful family name. Once the Julio-Claudians were finished, it was a title.
It seems overwhelmingly likely that when later emperors kept using it, they were referencing the dynasty that Augustus founded, not Julius Caesar personally.