Would it change the context of the sentence if I say "OK, what's your story?"
It would change it a bit, yes! So in this context, "what's your background?" is the same as asking about the person's experience with flying. Whereas, "what's your story?" would be more general, and the answer could include things not related to flying and airplanes. Ex: So, what's the story behind you getting a cat? Ex: Welcome to the job interview. Tell me, what's your background in engineering?
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