You don't have to prove much of anything to register a vehicle. The registration permits taxation, periodic smog checks, and verification of insurance.
Insurance companies issue small certificates, one copy of which is to be in the vehicle and one copy is submitted with registration renewals. There is no requirement that a person be a citizen; just a Kalifornia address would do.
Once the registration has been filed, the insured can probably cancel the insurance, receiving a pro-rated refund, and there is no requirement that the insurance company inform the state. If there is a lien-holder listed as "legal owner" and loss-payee, then the insurance company would notify them of the insurance cancellation.
If the state is not willing to jail or seek deportation for people who are not in compliance with these laws, then there is little motivation for them to be in compliance. I think that most of these laws are overly burdensome and should be repealed. But I resent the fact that they can be enforced against the law-abiding and yet there are no significant consequences for illegals.