I know the common usage of the word rider is for motorcycles and bicycles, but I would like to know if you can call a person who drives a car a rider?
In a car, a rider is usually a passenger, not the driver.
Rider — Vocabulary
n. 1.a. a traveler who actively rides an animal (as a horse or camel)
n. 1.b. a traveler who actively rides a vehicle (as a bicycle or motorcycle)
n. 1.c. a traveler riding in a vehicle (a boat or bus or car or plane or train etc) who is not operating it
I have to say that I have never come across rider when driver or passenger would be the accepted and common usage to cover a car. However, your question is interesting as I consider it perfectly correct to say, "He [neighbour] gave me a ride to work in his new car". To my way of thinking I remain a passenger in my neighbour's car and not a rider in this situation or context. . I wonder if we have a clash of American English versus British English here? In the UK it is common to ask someone if they would like a lift to work and this would be readily understood whether the mode of transport is a car or a motorcycle. However, I don't think lift would be understood in American English in this context whereas ride to work would be. But then the situation becomes even more complicated because we talk about a pillion passenger (BE) in the context of a person who sits behind the rider (the person in control) of a motorcycle.
Pillion passenger: A pillion is a mostly British English term for a...seat behind the main seat...[of a] motorcycle...A passenger (my emphasis) in this seat is said to "ride pillion"... (Wikipedia)
Definitely not! The person who drives the car is the "driver". Any other people travelling in the car are "passengers". Neither would be referred to as "rider" in UK or US English.
US English would possibly use "rider" in the context of a passenger in a train or bus, but these would remain "passenger" in the UK.