Upon the filing of such application, the department shall, upon registration, assign to such motor vehicle or trailer a distinctive registration number in the form of a license plate. Beginning on the implementation date designated by the director pursuant to subsection (4) of section 60-1508, for trailers which are not required to have a certificate of title under section 60-137 and which have an identification number issued by the county treasurer or department under section 60-386, trailer identification tags shall be supplied by the department and shall be required to be affixed to the trailer after issuance. Upon sale or transfer of any such motor vehicle or trailer, such number may be canceled or may be reassigned to another motor vehicle or trailer, at the option of the department, subject to the provisions of the Motor Vehicle Registration Act.