It is a cool sounding propaganda term, they certainly did know their propaganda stuff. I honestly don't know how that "storm" naming came to be but in German you you have the word "stürmen", which does translate quite directly if I am not mistaken, as in "storming Congress". I heard Maga knows a thing or two about it.
Now that I think of it I am not sure they were named after a storm. "stürmen" means moving rapidly into something and "erstürmen" means conquering/taking hold of something. "Sturm" is merely the noun for that. The weather phenomenon is rather named after that, than the other way round.
There is also the cultural epoque of "Sturm und Drang" btw (which had nothing to do with Nazis and predates them, emphasizing emotion over rationality in art in the late 18th century), which is also not about storms but about pressing forward. Maybe taking hold of something.