Did they call him "Octavian" in the first season, i.e. before the opening of Caesar's testament (which included announcement of the adoption? In which case, yes, inaccurate, although they may conceivably have decided to do it not to confuse viewers more (I personally disagree; the viewers who pay attention would have known, and it would have been easy to slip in a line along the lines of "Have you seen Gaius Octavius?" "Oh, didn't you know? He's Gaius Julius Caesar Octavianus now." "You don't say?" etc. etc.=

Okay, they're not hiring me any time soon as an extra writer...