The Fashion Report

[Yue_XIV]

there are 8 families, black, white, and the next six circles on the dye menu (last one doesn't count).

I believe grey is a separate family too. The pattern I've noticed is that the families follow the color of the paint bucket in the item icon. And yeah, the last tab containing the general-purpose dyes isn't its own family, instead every dye in there is part of one of the other families.