No motor vehicle designed to carry passengers, owned or leased by the state, or any of its departments, bureaus, commissions, or institutions supported in whole or in part by funds provided by the state, shall be operated or driven by any person unless it has displayed, in a prominent position on the rear of the vehicle, a license plate that shall be the same size, shape, and treated for increased visibility in the same manner as those issued by the registrar of motor vehicles for private vehicles. The license plate shall be attached to the vehicle in the same manner as provided by statute for the illumination and attachment of a license plate on private vehicles. The registrar shall designate the colors of the license plate that shall be used on state-owned cars; the colors shall be other than those used on privately owned motor vehicles, and shall apply only to license plates used on state-owned motor vehicles. The plate shall display a special serial number, and the words “Ohio State Car.”