From genius.com: "Genius Media Group, Inc. (GMG) is fully licensed to display lyrics across all of its properties. In 2013, GMG entered into licenses with every major music publisher: Sony/ATV Music Publishing, EMI Music Publishing, Universal Music Publishing Group, and Warner/Chappell Music. In addition, GMG developed a form license with the National Music Publishers' Association (NMPA) which today covers more than 96% of the independent publisher market."

Original copyright holder could give someone else authorisation to sue on their behalf, e.g., through an assignment. Doubtful Genius got an assignment in the agreements they have with publishers.

Also, Google claimed it is sub-licensed to re-publish through a third party, LyricFind, which has licenses with "over 4000" music publishers.