Copyright and trademark are two entirely different things.

Copyright protects the right of authors to decide how their work is used -- it applies to the content, e.g. the code.

Trademark protects the right of consumers to not be misled by fakes or frauds -- it applies to the names and identifiers that people apply to products and services, e.g the brand name.

Open source copyright licenses allow you to use the source code, but they typically do not grant any trademark rights.