"What's your hobby?" is phrased as if it can be assumed that the person definitely has one hobby, and only one hobby. You could ask "What are your hobbies?" or "What hobbies do you have?" A question about hobbies is about deliberate leisure activities that you are quite dedicated to. Most people would not list watching television or films or listening to music among their "hobbies", even if those things are frequently part of their leisure activities. A person with a deep interest in cinema might however list films (movies) as a hobby. More obvious examples of a hobby could be fishing, going to dance classes, stamp-collecting or karate. "What do you do in your free time?" is a more general question, so covers hobbies but also more casual and haphazard activities such as watching TV.

‘Hobby’ has a connotation of intellectual pleasure so it doesn’t (precisely) cover things like sports, physical activity, pure entertainment etc. which are some of the most common things that people do. I don’t think that most Americans consider any non-working time, such as doing chores ‘free’. In my experience, ‘free-time’ implies enjoyment or personal benefit.

No. At least not in my life experience. There's not much reason why you can't use both to mean the same thing. Is there a difference between the two? Sure, something you do in your free time like eating food may not be a hobby. "What do you do in your free time" could mean hobbies, but also life needed tasks chores like shopping, cleaning, hygiene, caring for children or husbands/wives, ect. While "What's your hobby" is explicit, and is almost always done in free time. In observed at least American talking culture, what people mean when they ask, "What do you do in your free time" they very likely mean "What do you do for a hobby", because most people (I would think) would not treat chores as hobbies. Unless you really are that interested in cleaning.

No that’s not true. We ask occasionally what is your hobby to see what they like. Now some people might ask what do you do in your free time if you don’t know what a hobby is 😂 However both is used and saying one or the other would not be in correct! Great job!