This is a bit of an odd ask, I know.

I'm looking for something along the lines of Black Haze, Darker than Black, or Code Geass, where the main character has a [famous] secret identity with superpowers.

Pretty much any genre is fine, just no comedy or shounen. I cannot stand either of them. Seinen is fine, though. Thanks!

You sure you don't want Shounen? Even Code Geass is shounen. 🤣

I'll throw one out there.

Sword Art Online - People are trapped in a virtual reality death game match. The main character hides his level, wears a newb outfit, and plenty don't recognize he's the infamous "Black Swordsman."

Hmm I wasn't able to watch farther than the first episode with the anime (because you had to pay after that) but Ghost Hunt was a pretty good supernatural manga. Once the plot thickens, you are slightly left wondering just who exactly is the main guy?

The only thing I can think is Mahouka Koukou no Rettousei. I also second ghost hunt since its a real good anime.

Okay, why is everyone recommending Ghost Hunt? It is definitely not a supernatural action anime/manga a la Superman that the OP is looking for.

As for my own rec's: they're both anime, but Samurai Flamenco and Tiger & Bunny fit pretty well.

lol sorry calstine ^^; I think I just saw one of the criterias as supernatural and went ahead with whatever came up to the top of my head.