It was technically a Bellamy salute (https://en.wikipedia.org/wiki/Bellamy_salute), which the Americans invented and used a lot before WW2. Not that this excuses anything, it is still out of style.
No need to get defensive! Lots of things that are causally unrelated to WW2 happened before it, and there are many people around trying to excuse Musk's Nazi salute as something that it is not, including as a "Bellamy Salute" or "Roman Salute"
I wouldn't be surprised if he sees himself as some sort of Wernher von Braun.