The Waipio Valley road, as I understand it, is mostly paved but extremely steep and narrow/twisty. So the 4x4 requirement is probably more about being able to control speed without burning up brakes and dealing with often wet conditions.

However, in the general case throughout most of the US West, "4WD roads" do indeed tend to be as much about having high clearance as about having 4WD per se. It's also just a way of saying that, if you just have a random rental car and aren't used to driving very rough and often narrow/exposed roads, this is probably not a good idea.