No, Musk had literally nothing to do with the name, just a continuation of his myth as anything more than a guy with a lot of money.

He started as a guy without money and got to briefly be the guy with the most money which is not so easy to do.

Are you sure he had no money? Was he born in a low or middle-class family?

I think Bezos has demonstrated quite convincingly that simply having a lot of money does not mean you can compete with SpaceX. Similarly, I don't know much about the automotive industry, but turning a company with a few employees into one of the most valuable companies in the world is not easy, even if you have $100 million to play with at the start.

It's OK to simultaneously believe that "Elon Musk is very effective at growing engineering businesses to the point that they disrupt entire industries" and also that "Elon Musk is not as smart as he thinks he is, and his tweets are embarrassing".

Is it really a significant part of his myth that he named the company?

I didn't say it was significant! Though, being considered a 'co-founder' of Tesla was important enough to him to sue the actual founders.

"Tesla almost had a very different name—until Elon Musk and his co-founders bought the rights to “Tesla” for $75,000"

https://fortune.com/2018/12/09/tesla-name-faraday-elon-musk/