Answering your questions about this particular expression. I’m glad you both asked about it.
snavej said: Doesn’t doko mean where?
Yes, you’re right. Doko does mean where; however, in this context it means “why.” So another good translation would be “Why do you like him/her?” Then, you would list your reasons.
Another interesting expression related to this is 私のどんなところが好きですか (Watashi no donna tokoro ga suki desu ka) “What kinds of things do you like about me?/What do you like about me?” While ところ means place, it also refers to abstract concepts, things, points, and characteristics.
bellfry said: Why does the structure for あなたは彼女・彼のどこが好きですか employ “where” instead of “what”? Is it usual usage?
As for why doko is preferred over nani or even doushite, I didn’t really get a definitive answer other than doko references personality, characteristics, traits, etc. It’s just one of those expressions that don’t have a definite translation into English–something you just have to accept. But, this does seem to be a typical expression that is often used. You could actually use this pattern for other things, too.
What do you like about me?
What do you like about Japan?