Not necessarily. In Britain, "a motor" can be an automobile whether it is powered by gasoline or electricity and "an engine" can be a locomotive whether it is powered by steam, diesel, or electricity.
However, I would like to know if "motore" is only the power plant or can it be the whole vehicle?
Motore only means "engine" or "motor".
Car is "macchina" (or "auto"/"automobile").
Some translations equate "il motore" as "the car" but they are wrong - it is actually the "engine" they are referring to: e.g.
Vado a scaldare il motore dell'auto. (I'm going to warm up the car.) Il motore originale era da 5,5 litri. (The original car has 5.5 liters.)
See my comment above. While some translations of motore give car, it is really the engine they are talking about. For example: Abbiamo problemi al motore. We are having car problems.
"Car" is just a shortcut for where the real problem lies.
Motore is the motor or engine.