Roman names are quite complicated and the name of a free borne citizen would (basically) be comprised of five parts: 1) Praenomen (first name), 2) nomen/gentilicium (family name), 3) cognomen (branch of family), 4) fathers name and 5) indication of the Roman voting tribe. Women did not have a praenomen after an early period in Roman history, or voting tribe. Normally they just received a feminine form of the nomen (Gaius Julius Caesar’s daughter was names Julia). Subsequent daughters were often just called second, third etc.
For men:
Tthe praenomen is very repetitive in Roman culture and only used by intimates – in the Late republic only 18 praenomina were in general use: Aulus, Decimus, Gaius, Gnaeus, Lucius, Manius, Marcus, Numerius, Publius, Quintus, Servius, Sextus, Spurius, Tiberius, Titus, and the Oscan Salvius, Statius, Vibius. This name was chosen by the parents, often following the father, grandfather or other relative (Gaius in your question).
The nomen/gentilicium was the hereditary family name (often including several branches of a family). That is Julius in your question. This name depended on your father’s. The nomen of a master was adopted by slaves. A freed slave of Caesar would therefore be called Gaius (after Gaius) Julius (after Julius) G. (short for Gaius) l(ibertus) [slave name]. The same general principle applies to individuals becoming Roman citizens. An effect of this was that a lot of people ended up carrying the family names of the imperial families.
The cognomen (Caesar in your question) was originally an unofficial surname for individuals which became hereditary for branches of a large family. With time some families acquired several cognomina. By the imperial period if was generally established in the population. The cognomen was not given name (and could not be taken), but followed that of your family branch. Your problem could therefore not occur in practice. It would in one sense be like adopting someones Social security number/personal number/what ever ID your country use.
TLDR; you could not take the name Caesar because it was a family name, it’s simply not how Roman naming practice worked.
Reference: OCD^4 s.v. Names, personal, Roman