"Funny" is, in my estimation, the single most mis-used word by Korean learners of English. Somewhere along the line, they internalize a rule that tells them that "fun" and "funny" are synonyms, and that, furthermore, they describe an internal mental state rather than an external situation. Hence you get innumerable variations on this sort of phrase: "I am funny," by which is meant "I am having fun." I think the problem arises out of a semantic overlap in Korean that doesn't exist in English, but is aggravated by the deceptive shared etymology of the two English words.
Lately, I've taken to telling my students that their chances of using the word "funny" correctly are sufficiently low that their best bet it to avoid the word altogether. "Fun" has broader semantics in any event, and is generally closer to what they intend. The issue of the fact that it describes an external as opposed to internal state is more difficult to resolve, and is linked to Koreans' efforts to use English "state" adjectives in general, I think. Regardless, "I am fun" is slightly more comprehensible than "I am funny" as a multi-purpose response to an entertaining situation.