So, this is a question the answer to which could be something like "not very well," or "there is a pool in my neighborhood," right?
Như thế nào : it's impossible to tell in this case exactly what it means because the person could either be asking "how" (the method) or "how" (the quality). I do not think it could be understood as "how (where) do you swim around here?". If you want to specifically ask about the method, use làm sao bạn bơi? or làm sao để bơi?
I believe, the logical answer would be "Very well" or "Not very well". The answer about a pool would probably not work here.
"Làm thế nào để bơi?" - This would be "How do you do it, how to swim". It is the same word "how", but from my understanding, if you are just asked "How do you swim", then we are probably talking about the degree of skill. Though I may be wrong, of course.
If the word is the same, then wouldn't it depend on context, as in English? For instance, if my sentence before this was "I swim a lot in downtown Chicago," then I would assume the question was about instrumentality. If, on the other hand, I had just saId "I am planning on sailing across Lake Michigan," I would assume the questioner was asking about ability.
Of course the context would always be important. Both downtown of Chicago and a pool in the neighbourhood are not instruments, but locations. I talked to a Vietnamese native speaker and she told me that in this context a more appropriate question in a discourse between the Vietnamese would be "Where do you swim" and not "How do you swim". It does not work the same as in English and the question from the lesson would generally not be used to ask about what you call "instrumentality". Here the question is very general and is about the quality of swimming, the degree of your skill, not the specifics of how you do it.