No. In Texas, a licensed driver cannot purchase a standard liability auto insurance policy unless a vehicle is registered in their name.
Auto liability insurance follows the vehicle, not the driver. The policy is tied to a specific VIN. To insure a car, you must have an insurable interest—typically ownership (title/registration) or a lease.
If you regularly drive a specific car (e.g., a parent's), you should be listed as a driver on that car's policy.