You may be able to register a car in one state and insure it in another, it just depends upon the circumstances for doing so and the state laws involved. Some states require that if your car is registered in your state that you also only have it insured by a state licensed insurance company.
For example in New York State in order to register your car and obtain license plates in NYS you must either purchase an automobile liability insurance policy from an insurer licensed by the New York State Insurance Department.
For your example of Idaho and California see below.
To register a vehicle in Idaho the ID Department of Motor Vehicles (DMV) states that the vehicle must belong to an Idaho resident or business and the vehicle must be titled in Idaho before it can be registered. Idaho residents and businesses may title and register vehicles in Idaho. Idaho residents who are temporarily in another state (i.e., students or military personnel) may request titles and registrations for their vehicle(s) through the mail.
Idaho Statute 49-1229 requires motor vehicle insurance. Here it states (1) Every owner of a motor vehicle which is registered and operated in Idaho by the owner or with his permission shall continuously, except as provided in section 41-2516, Idaho Code, provide insurance against loss resulting from liability imposed by law for bodily injury or death or damage to property suffered by any person caused by maintenance or use of motor vehicles described therein in an amount not less than that required by section 49-117, Idaho Code, and shall demonstrate the existence of any other coverage required by this title or a certificate of self-insurance issued by the department pursuant to section 49-1224, Idaho Code, for each motor vehicle to be registered.
Idaho Code 49-117 (18)notes: "Proof of financial responsibility" means proof of ability to respond in damages for liability, on account of accidents occurring subsequent to the effective date of the proof, arising out of the ownership, maintenance or use of a motor vehicle, in the amount of twenty-five thousand dollars ($25,000) because of bodily injury to or death of one (1) person in any one (1) accident, and, subject to the limit for one (1) person, in the amount of fifty thousand dollars ($50,000) because of bodily injury to or death of two (2) or more persons in any one (1) accident, and in the amount of fifteen thousand dollars ($15,000) because of injury to or destruction of property of others in any one (1) accident. So Idaho limits are 25/50/15. It does not list that your insurance must be purchased in Idaho however.
California's minimum requirements are Bodily Injury Liability coverage of $15,000 per injured person up to a total of $30,000 per accident, and Property Damage Liability coverage with a minimum limit of $5,000. This basic coverage is often referred to as 15/30/5 coverage and is less than Idaho's auto insurance limits.
If Idaho does not require you to have your auto insurance purchased through an ID insurer then it may be possible to insure the car in California if you are currently residing in CA as a non-resident that does not need to register the car there. For instance if your permanent address is in Idaho but you are a college student in California your Idaho insurance carrier may not cover you to be in California since the car is garaged in CA a majority of the year. You then may be able to obtain California insurance however you will likely need to get limits of 25/50/15 to meet Idaho's minimum requirements since your vehicle is registered in that state.
If you have moved to California and taken a job, etc thus are not a student or in the military and instead are now a full time resident of CA you will need to register and insure the car there instead of keeping the car registered in Idaho. Registration fees must be paid within 20 days of entry into CA or residency to avoid penalties.
So if you want to have your car registered in Idaho and insured in California you will need check with the ID DMV first to make certain it is okay then find an insurance company that would provide you this type of insurance policy.