The difference is that the left is dominant in the relevant bureaucracies and professional fields. Higher ed administration, HR staff, public school teachers and administrators, journalists, tech companies, pretty much all of them lean left. So when they draw their red lines and police social interactions, they're doing so on average more so from a left POV.
The right wants to resist this because they don't like the newfound cultural dominance that the left has achieved through its so called "long march" through the institutions combined with recent mass mobilization. But they're trying to do so in a clunky, top down way that actually is closer to proper censorship and the chilling of speech by the government with these anti-CRT and anti-protest bills.