He has Degrees in physics, and many people with such degrees end up getting positions in engineering companies and are called engineers. He was accepted at Standford PHd program for materials science. That would natural lead into an Materials engineering position in some company if completed.
Of course instead of doing that, he left and started companies, as a software engineer he was the core developer one Zip2 and early on on PayPal. If that doesn't qualify you as an engineer I don't know what does.
Literally he has been the Chief Engineer at SpaceX since its founding. And the argument that its just an empty title just don't work. They were a small company for quite a while and somebody made all the final decisions. The are multiple stories where he works directly with other engineers on the rocket as well if 'making engineering decisions' is not enough. Multiple former employees have directly contradicted these sort of 'not engineer claim'.
Just recently former lead engineer at Ford with 50 years experience in automotive, defense and aerospace was allowed to sit in on a meeting for Starship. He said he had never seen anybody CEO be that involved in detailed decisions, about manufacturing processes and design choices and so on.
So just because you don't like him or his product, doesn't make him not an engineer. He has the education many other people that call themselves engineers have, and he has the title of engineer and that title has been officially recognized by NASA and DoD.
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Elon is both the Chief Executive Officer and Chief Technology Officer of SpaceX, so of course he does more than just ‘some very technical work’. He is integrally involved in the actual design and engineering of the rocket, and at least touches every other aspect of the business (but I would say the former takes up much more of his mental real estate). Elon is an engineer at heart, and that’s where and how he works best.
Former Head of Software Quality Assurance at SpaceX