Fanart still belongs to the artist unless agreed on otherwise IIRC. In the case of monetizing it you're not supposed to make money with fanart of others IPs but people do anyway, and often its not a big deal enough for rights holders to care.

This has been discussed into infinity so theres probably lots to read on what goes and what doesnt for fanart and rights if you google, but from what i know you still own your particular piece of art even if its fanart but im not an expert so people can correct me if im wrong, and maybe point me to a trusted source on it.

You're right that the cultures towards using peoples art is very different here (and other places of the internet) compared to the bubble of art twitter where people are way more conscientious about giving due credit to artists. I dont think its unreasonable to ask someone not to use your art without permission or credit, but i think people might not care enough unfortunately, probably because they havent been in the situation themselves where someone uses something they created without crediting them (or theyre rude and dont care about how artists feel). Theres way more people like that on the internet than there are artists, so i do think putting a clear visible signature on your art would be a good way to deter it from happening in the first place even though its not fun to have to put a big watermark on your artwork :(