So a war that ended almost 80 years ago is responsible for the battle against "online hate" that started like two years ago?

Contrary to popular belief, human culture didn't begin on Twitter. The battle against "online hate" is just the current generational manifestation of a conflict that, yes, has its origins in the ideologies and cultural shocks of World War 2. Particularly where the meme-complexes of modern anti-Semitism and fear of "cultural Marxism" in academia, gender roles and homophobia are concerned, many of which were established either by the Nazis directly or the initial right-wing political shift in the US following the war. Today's redpillers, anti-wokists and the like are just rehashing beliefs that were old hat in their grandparents' time. Even the culture war over transgender identity isn't new - there was an episode of All in the Family about it in the 1970s.

I mean go look up any of George Carlin's old bits on political correctness and they would fit in perfectly in a modern day context, despite being made before the WWW was so much as a gleam in Tim Berners' Lee's eye. You'd just have to update the technological referents.