LT Sackett is correct on using a car as a midshipman. However, there is likely to be lots of opinions on insurance and licenses.
You should be able to keep your mid on your insurance just as you likely would for any college. He can also use his own company and most mids choose USAA given their outstanding ratings and military mindness.
The license is a portion of a much larger issue...home of record. Most states allow you to keep your license because you are "ordered" into the military. For example, I entered the military from NH and kept my license in NH for over 35 years; but that was because I chose to keep my residence in NH for a variety of other reasons, the biggest being no state income tax. Although LT Sackett is likely correct in saying that the mid can probably get a MD license, the question needs to be whether that is correct for all the right reasons. When it comes time to "prove" a residence, having a license from the state is a good one, so if it is beneficial to keep a "residence" in CA, so be it, but you will find many members of the Armed Forces who are "residents" of TX, FL (remember 2000 election!), NH, WY, AK...all because they have no state income tax.