In Death Valley they often specify 4x4 high clearance. Of course, you don't REALLY need 4wd or even 2wd; you can tackle it on a motorcycle :)
In Death Valley you rather need more spare tires (if you go e.g. to Racetrack; seen a car that got 2 punctures at different locations). Otherwise high clearance is needed only if you go off-the-beaten track; even Titus Canyon is fine on a normal road vehicle.
And in particular, the Hawaiian rental car counters will tell you when you rent which roads your car is not allowed to go on.
I've always wondered why this matters. I've got car insurance, I've got additional rental coverage on my credit card, and there are plenty of other vendors from whom to rent cars if you want to blacklist me.
Having violated this all the time I've been to Hawai'i I can understand why do they do that - for example, if you continue on the road to Hāna/Maui beyond, down to the southern side of Haleakalā the road becomes really dodgy, later turns into gravel and there is no water/gas for xy miles available, so for casual American tourists it might be pretty scary. Also, driving the north-western part of Maui is on a very narrow road over cliffs and fallen rocks (not as bad as parts of Tenerife though). I actually had an encounter on a pretty tall cliff where I was driving Subaru Forrester and some US female celebrity named Kim was driving the opposite way in even a larger car; we drove past each other with smiles in about 1mph to avoid collision and she must have been right on the edge with her car while I was almost scratching the right side rocks. Even Saddle Road between Maunas on the Big Island could be pretty demanding to casual tourists in fog and really slippery when descending down to Hilo in strong rain, which is like always (saw a crash once). There are also some scary roads on Kaua'i and O'ahu. In US it's so much easier to get a driving license than in EU, so your casual drivers are not used to fast and safe driving like in Germany, many off-road segments like in Spain or navigating chaotic situations as in 3rd world countries, and your society is generally on the safe side (even excessively so).
They probably don't want you taking a road you shouldn't, getting hurt, and then suing them because they didn't warn you about the dangers. Or they get tired of having all their cars smashed up. Insurance covers it, but it's still annoying.
Well, they still have trouble with you if you get stuck or wreck the car, so it is easier for them to just list the "forbidden" roads. Not sure, but your insurance might be void if you crash the car where you were not supposed to drive.
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.