You can have a copyrighted derivative of a copyrighted subject - for example, buying a stock photo of the eiffel tower at night pays both the photographer and the lighting designer; and buying a translated book pays both the original author and the translator. This is known as a "derivative work".

However, to be copyrightable as a derivative, work must contain sufficient new expression to satisfy copyright law's requirement of originality. So you can't copyright a photocopy of a book, for example.

Presumably Genius thinks their lyrics are analogous to the photograph, rather than the photocopy; they may not own the lyrics, but they own this particular transcription of the lyrics.