I think that in response to the question, "How do you like your coffee?" the natural response is "I take my coffee black." It would sound odd to say "I drink black coffee" in response to such a question, because it sounds like you're talking about something you do habitually - so, not really answering the question - instead of stating your preference.
If the question were, "Do you drink black coffee?", then "I (do) drink black coffee." would make sense, because the person is inquiring about whether you drink black coffee or not.