The word car was already common long prior to the invention of the automobile. It originally was another word for carriage or cart, from the french carr. It seems that it was a sort of generic word for the part of a horse-drawn conveyance that one rode in, as in "I'll hire a horse and car."

In the 19th century, the definition of the word car expanded to encompass the parts of other kinds of conveyances that people rode in, such as the basket of a balloon or airship or the passenger compartment of an elevator. It also became a common descriptor of railroad cars, including streetcars. Finally, to the street car and railroad car were joined the motor car, which was shortened to car and became the car we all know. (Main source: Oxford English Dictionary)