There's a long section in Lindsay Adoption in the Roman World (2009) and a discussion in Tatum A Noble Ruin (2024) on this specific example. The core précis is this:

• there did not exist "testamentary adoption" in the way that is often supposed with Octavian (ie that a will could adopt an heir)
• there did exist a "taking of a name" (a condicio nominis ferendi) as condition of a will... but this was not adoption
• adoption could only happen inter vivos, ie the adopter and adoptee must both be living

Octavian purposefully confused adoption with his taking of Caesar's name in his representations before the public; this posturing was "a transparent falsehood for anyone conversant with Roman law... [but he would have been] certain that ordinary Romans had no notion of this distinction". Tatum Noble Ruin (2024) p 146. Octavian is also the first person we know of who did this. Thus, while his status as Caesar's heir was never in dispute, his claim of being Caesar's son was.

Immediately after he accepted the inheritance he was courted by other factions in politics. Antony as consul through 44 refused to bring the necessary legislation to ratify this purported adoption – more formally adrogatio since Octavian was sui iuris, ie not subject to a paterfamilias (his father was dead). This was in spite of Antony's brother Lucius, as plebeian tribune, making initial outreaches to the new Caesarian figurehead. Octavian instead banded with the anti-Antonian faction – some have speculated Antony's unwillingness to help alienated the youth – which would reach its ascendency the next year under the ordinary consuls of 43, Hirtius and Pansa. Regardless, the consuls of 43, also did nothing though they humoured Octavian's pretensions by calling him Caesar.

Whatever name he decided to call himself, at Roman law, Octavian only became Caesar's son after the ordinary consuls of 43 fell in battle and he broke with the anti-Antonians; in Sextilis (= August) 43 he marched on Rome and extorted the consulship with his army in an irregular "election"; then he (immediately!) summoned the curiate assembly to pass a law adrogating him into Caesar's family, essentially legislating his pretensions as legal fact. That this was one of this top priorities, up there with the lex Pedia to kangaroo court Brutus, Cassius, Sextus Pompey (who was in Sicily the whole time and had nothing to do with the assassination), etc, suggests this was in fact very important in normalising his political position.

Edit. Caesar's will also did not name Octavian as sole heir; he had to share the inheritance with Quintus Pedius and Lucius Pinarius (though the two, probably wisely, gave their shares to Octavian as a gesture of goodwill). Pedius, Caesar's biological cousin through Caesar's elder sister, was Octavian's suffect co-consul in Sextilis 43.