While Disney+ and DC Universe/The CW both have their own Stargirl - but the Disney+ Stargirl movie and the DC Universe Stargirl TV series are completely different. This has caused some marketing confusion, with the Disney+ movie releasing as marketing for the TV show has been ramping up.
The Disney+ Stargirl movie is an adaptation of the Stargirl young adult novel released in 2000, written by Jerry Spinelli. The DC Universe Stargirl series is based on the character Courtney Whitmore, aka Stargirl, created by Geoff Johns and Leo Moder for DC Comics in 1999. They come from completely different places, and that’s the key to understanding why they both can be made at the same time by different companies.
Disney+ and DC Universe can both make something called Stargirl because they’re so different that the trademarks don’t overlap. A trademark is designed to be highly specific, not just a blanket monopoly on using the term anywhere. Outside of the occasional comic adaptation of a novel or young adult novel starring comic characters, there’s not generally a lot of crossover between the two. Add in just how different the stories themselves are, and there’s no reason one trademark should get in the way of the other.
The Stargirl TV series is based on the DC Comics character. She’s connected to the Justice Society of America, taking the staff left by Starman to become a superhero of her own. There have been versions of this character on TV recently, appearing in Legends of Tomorrow season 2 as part of that season’s Justice Society of America arc. The new version appeared in the final episode of Crisis on Infinite Earths, taking place on the Arrowverse's new Earth-2 with a new Justice Society.
The Stargirl movie is based on the young adult coming-of-age story. The title character is Susan “Stargirl” Caraway, a high school student who is different than her peers, even by the standards of previously home-schooled students. The movie, like the book, is set in a very normal world; it’s just the title character who’s "weird", learning that she’s better off being weird rather than trying to adapt to the world around her.
Any trademark on the name “Stargirl” would consider these two stories very separate. There’s not a lot of overlap between a young adult coming of age novel and a DC comics superhero, so separate trademarks would be granted for them with little difficulty. The only reason there’s any confusion right now is that they’re coming out so close to each other. Had the Stargirl movie not gone through so many development issues, it would have come out a few years ago and the similar names would be a trivial note, rather than a cause for confusion. As they’re very different stories sharing nothing more than a name, there’s nothing to stop Disney+ and DC Universe both putting out something named “Stargirl”, even at the same time.
