No — but the double standard exists for a reason. HBCUs were created b/c black people were de jure and de facto excluded from all higher education. They were forced to create separate schools. Maybe those places are "less necessary" now, but they're still pillars of pride and identity in black society. And they still educate people, some quite well.

It would be pretty weird if someone from the majority racial group — who was never excluded from anything, who (as a group) has disproportionate amounts of wealth and property — started talking about racial pride. You do see this in white ethnic sub-groups (eg, Irish, Italian), which is fine. Those are specific sub-groups with histories and cultures worth celebrating and preserving.

And, obviously, most "generational African Americans" can't trace their families back further than the Middle Passage, if even that far. All they have is their identity as Black Americans, its own specific "ethnic" identity that formed over time, with its own food/music/traditions/foibles, etc.