It was embraced by the left first. The right just picked up on that, picked out the most extreme and questionable examples, then exaggerated those, and used it as a weapon to own the libs.

But, prior to right-wing distortion, it absolutely was not just "respect for humans and human rights". It had far more specific of a meaning than that while the left was using it before the right realized what a wonderful PR weapon the term would be. For instance, a particular flavor of anti-racism (sans empirical evidence that it's effective, and don't you dare try to argue that that flavor itself promotes certain types of injustices) has always been part of what it means to be woke.

Furthermore, even before the right distorted the meaning, the woke left would do the equivalent of proclaiming 'love thy neighbor' while shaming or burning at the stake the heretics who questioned their motives and the soundness of their judgments.

Very evangelical in character.

(I don't really think Nietzsche is the best perspective from which to understand the phenomenon, though.)