Although the word “car” precedes the word “carriage”, both have similar roots. I know that the term “motor carriage” was used in the early days of the automobile. It would be my assumption then that this was shortened to “motorcar” and then again to “car.” If that assumption is true, would that then suggest that “car” as it refers to an automobile is short for “carriage” or were the terms “motor carriage” and “car” (as derived from the French for “carre”) used separately to describe an automobile?
“Carriage,” when adopted into English, was the noun-form of “carry”; it denoted an act of carrying, and to describe various things associated with that (such as a fee). Its usage for a horse-drawn coach dates back no further than to the mid-18th century, and for a wheeled conveyance in general to the 16th century.
“Car,” on the other hand, while it may have arrived in English via Old French and Latin, has meant “a conveyance (usually wheeled)” since the Romans adopted it from Old Celtic. Latin carrus is the ancestor of “car,” “carry,” and French carre.
Whether a particular automobile was named a carriage, a car, a coach (16th century, ultimately from Magyar, of all things) appears to have been at the whim of inventors, early adopters, and the like. I suspect the choice depends, in part, on what other things were described by those words at the particular place and time. The great age of development for automobiles fell squarely in the days of “carriage means ‘a well-sprung wheeled conveyance for carrying passengers, drawn by a horse or horses,’” whereas “car” mostly meant “vehicle that carries freight or passengers as part of a larger mechanism, such as a railway, tram, balloon, etc,” so it’s no surprise that “carriage” was the winner. I’m not sure anyone knows how “motor car” came to be adopted, but while “it’s shorter” seems likely to me, it’s not the case that “car” is a shortening of “carriage.”