The voices of Cinderella and her fairy godmother are imitations of Bette Davis and Barbara Jo Allen (aka Vera Vague), respectively.
Audiences at the time would have noticed the "A" gasoline rationing sticker on Granny's scooter. This was the most common level of rationing issued to the general public. It allowed the holder to purchase four gallons of gasoline per week.
The fairy godmother takes Cinderella's call at Duffy's Cavern, a pun on the radio program and film Duffy's Tavern (1945).
The fairy godmother's second mode of transport (created from a trash can) is a US Army jeep, familiar to wartime audiences.