Can "drive someone home" be used when the vehicle is a motorbike? Can I use "ride someone home" with the same meaning as "drive someone home"?
Well the original definition of "to drive" was to compel something to move by force.
For instance, the phrase "I drive my friend" could have an entirely different connotation if you add "to succeed in sports". Now you're driving him morally, not within a car.
The word drive is, in this way, very flexible. The original definitions extends to metaphors and analogies, meaning any time you're influencing something towards a certain ends you can consider it driving.
Therefore, yes- you can grammatically say you "drive you friend" on a motorbike.
The reason it sounds strange is because any time you're talking about a motorbike you expect to hear the word "ride", not "drive". This is simply a common cultural pattern. In your specific case, however, saying you "ride someone home" is not the same as "drive someone home".
"To ride" is to be in or on something that is moving. Thus, if you "ride your friend home", you're on your friend's back and riding him like a horse =)
If you want to use the word ride, try:
"I gave my friend a ride home", or "I rode my friend home on my bike".
Note that the second is acceptable because the preposition "on" clearly identifies the direct object (the object being ridden), which leaves the other noun ("my friend") to be the indirect object - one who is affected by the act of you riding the bike.