(a) Under this subchapter, the ordinance of the governing body of a city or town may contain therein a classification of vehicles by types and the rate of tax levy, stated in dollars and cents, to be collected from the owners of the vehicles coming within these classifications.
(b) No such classification shall, at the time of the adoption of the ordinance, include any vehicle for the use of which a state tax or fee for the registration or licensing of motor vehicles is not, at the time, levied upon the owner.