Since you're in California, it's basically what Mark said. Here's the quote from the state's website:
Generally, every sail-powered vessel over eight feet in length and every motor-driven vessel (regardless of length) that is not documented by the U.S. Coast Guard which is used or on the waters of this state are subject to registration by the Department of Motor Vehicles (DMV). The vessel must be located in California.
The exception in California is a documented vessel, which of course your dingy isn't.
Every state is different. Florida it is:
Vessels exempt from registration include: non-motor powered vessel less than 16 feet in length, and any non-powered canoe, kayak, racing shell, or rowing scull, regardless of length; vessels used exclusively on private lakes
and ponds;
So Florida is similar to California but slight differences. Also Florida does require registration of documented vessels.
Now I also agree with the other advice on registration in those states that have "tender exemptions." Sometime, somewhere, not registering will come to haunt you. For instance, bring that tender into Florida. Well, Florida exempts boats registered in other states and in Florida less than 90 days. So, you would not be exempt and would be required to register it in Florida. Although minor on a dingy, that would also mean paying use taxes.