Sometimes driving is used when the speaker is not actually the person behind the wheel. You can say "We are driving all the way to San Francisco to visit my son." even though 2 people can't drive a car at the same time. It is used to differentiate the method of travel in this case, not the performer of the action. The same way I could say "I am flying to Spain next week." Of course, I am not flying the plane, nor am I flying like a bird, I am just a passenger. Especially in the not too distant past, "riding" meant on horseback or by horse and carriage.

You understand the verb to drive very well. Your explanation is very good, almost like a legal definition. IN this context please don't drop me home is related to driving. In English dropping off someone means you drive them somewhere. So the passenger is saying please don't drive me home. Drop off is another way of saying that. Just saying drop me home is a little strange, but maybe that's just this song. It would be more normal to say please don't drop me off at my house/place if you were talking to someone driving you.