No, there may be inequality and bigotry in some solarpunk fiction but unlike cyberpunk it's not about "our heroes fighting the system that will almost inevitably crush them". Solarpunk is innately hopeful, and there's conflict (kinda intrinsic to storytelling) but it doesn't require the existence of inequality or bigotry, and a lot of solarpunk fiction explicitly doesn't have any bigotry in it period.

Cyberpunk might be about "our system sucks, and our heroes may or may not want it to change", but solarpunk is about "the system of the modern day was bad, and so we replaced it entirely". The "punk" part doesn't require that the heroes are individually punks within the context of their own world, it's called punk because it's in contrast to our modern system. Also because -punk is kinda a generic term for genres at this point.