Eh, not exactly. Trademarking has its own formal process, but copyright is a thing that just exists by default. If you make it, you own it. The exception to this is if you're producing work for someone else and they have you sign a work-for-hire contract, which transfers that default copyright over to them. You can safely assume that any (reputable) corpo owns all their character designs, and indie designs will belong to either the artist or the talent, depending on the arrangement between the two parties.
That's why the AI art situation is a bit murky legally to start with - to the best of my knowledge, there's no court precedent for it to establish exactly how transformative an AI-generated piece needs to be before it's considered legally distinct from the source pieces it pulls from. If it turns out someone's AI artwork is noticeably similar to an existing piece, the original artist might try to take action against it. They don't need to have registered anything anywhere, they can just say "hey, this looks like mine" and fire out DMCAs, starting a legal process.