If another driver did get into an accident with your vehicle, you’d want to ensure that it’s covered.
While they may have their own coverage, there may be an exclusion for this situation in their policy. If your vehicle isn’t listed on their policy, their insurance may not cover the damage. If the other driver is not listed on your policy, your insurance may also not cover the damage.
Your rate should not increase significantly if you add an additional driver. You can also comparison shop with other carriers to find the best rate with both you and your girlfriend listed on the policy. This will ensure your vehicle is protected in any situation.