The radio licence covered all wireless sets in your home - but only in your home. If you were lucky enough to have a car, and you were even more lucky to have a radio in it, then you were required to have a separate licence. In1938 the "conditions" on the back of the radio licence were changed to make it clear that car owners would have to pay out twice ... there is no evidence yet that this applied in earlier years.
Later on the design of the radio licence changed so that car registration numbers could be inserted onto the licence. Here are three examples of car radio licences from the Stephen Dunderdale collection
From 1959 (note car registration numbers)
and from 1969
In case you didn't realise you needed a licence for your car radio, posters reminded you !