There are laws on the books that require non commercial class A & B licenses on towing, length and gvwr. You don't need the physical but you are required to take both the written and driving tests.
As to the reason CA issues commercial plates to all pickup trucks is due to the fact that they can be used in commercial service, if you notice taxi cabs and courier vehicles all have commercial plates since they are used completely for commercial purposes. I don't know why they require personal trucks to sport commercial plates its has always been that way here. In AZ my pickup did not have a commercial plate since I don't use it at all in a commercial capacity. They did ask though before I got my plates.
I think anything over 40' requires at least a non commercial class A and any vehicle or trail with 3 or more axles also require one. I'm not sure where the gvwr kicks in to where you will need one but I'm sure it is related to number of axles and the length since you can carry more with those combinations.
You could petition the DMV to remove the commercial status of your truck to an RV and then you can get the regular plates. That's why you see some vans with non commercial plates and some with since they can be dual purpose.
