If a car is bought with cash (paid for in full) or if it is financed could be a rating factor that is applied to insurance carrier's rating algorithm, but we don’t know of any carrier that uses that question for that purpose.
The question of if a vehicle is owned fully or financed is asked because some insurance providers have filed with the state that Physical Damage coverage is required on a financed or leased car. Typically, it is the lien holder that requires physical damage, but some carriers have filed rates with the state that required physical damage on financed or leased vehicles. Esurance is one example of an insurance company that has this requirement.
Many insurance companies thus ask this question due to how they have filed their rates and/or to make sure that if the car is financed you carry Physical Damage since it is most likely required by your lien holder. Collision and Comprehensive are the type of physical damage coverage that is usually required if your vehicle is financed or leased. Lien holders normally require these coverages on a vehicle to protect their interest or collateral, the vehicle.