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GPL is an effective license for ensuring that your source code remains free and open source even when modified and released again by others. The fact that "Streamlabs OBS" is open source means that OBS Studio’s GPL license has fulfilled its purpose.

Trademarks are a separate matter. To protect its brand, OBS needs to register its name as a trademark. Having a closed source license isn’t enough to stop competitors from diluting the brand, and it would also prevent OBS from enjoying the benefits of having the free and open source software community contribute back to the project. If OBS trademarked its name, it would be able to threaten competitors like Streamlabs with legal action when they infringe on the trademark, regardless of whether the code is open source or closed source.