Even though the clans of Wano mean nothing to Kaido now, since he wants to establish New Onigashima. It's clear as day that he holds a grudge with the Kozuki clan to some capacity.
His dialogue throughout the arc is clear indicator of this not just being his enmity with Oden-
I wouldn't be surprised if the poneglyph encountered by Law in the Onigashima basement holds information about this Oni race and gives context to their history with Wano/Kozuki clan in the past. Really excited to see the bigger picture here.