Generally, I do all of my quotes online instead of dealing with an agent. They've always asked if the car is leased, financed, or owned. In my current situation I am technically financing the car, but the loan is unsecured. This means that the title doesn't have a lien on it.
So do I technically "own" or "finance" this car?
If the loan is unsecured, then you own the car.
If the loan was secured on the car, and you lost, sold or destroyed the car, you would have to repay the loan. Since it's not, you get to keep the debt until you pay it off, regardless of whether you still have the car or not.
You own the car.
The insurance company wants to know: leased, financed, or owned; so they know if they have to inform the "lender" that you have insurance, and let the lender know the amount of insurance. The lender wants to know that their collateral is being protected. The insurance company may need to contact the lender if you make a claim so that the lender knows that the car was damaged and repaired.
If the car isn't collateral, then your decision to insure the vehicle is up to you, though you still have to follow your state law regarding the other parts of auto insurance and amounts of coverage for things like liability.
This is about signaling.
A secured lender requires you carry insurance to protect them. The insurance company is asking who they are, to signal them that you do have insurance. And if you lapse, to signal to them that you do not, so they can take action.
Also, if totaled, the insurer can pay off the lender directly. (Consumers often can't be trusted to do this, not least they are desperate for a replacement car.)
Since you supposedly bought this on an unsecured line of credit, the bank doesn't even realize you bought a car with it. They would be quite surprised to hear from the insurer and wouldn't know what to do with the information.
You don't tell them. In fact if you do, they may assume the loan is secured and charge you extra for their effort, usually $50 to $100 a year.
(Not even if the loan is secured by other means, a guarantor, say).
In the event of a total loss, their duty of care may be to pay the true owner of the car, for the loss of their security, and the OP their equity (if any).
Some jursidictions have stamp duty on this extra fee, and there may even be a formal register of the security that is maintained, so clerical work all around.
