Prescriptivism, or descriptivism, that is the question.
Strang 1970 and Copperud 1970, 1980 note that there is conflicting usage with the first person pronoun: "It is I who (is? am?)." Strang points out that in earlier stages of the language such a sentence would have been unmistakably governed by the first person pronoun and would have begun "It am I...." But over the years the position of it caused it to be felt to be the true subject, and the third person verb is replaced am and sometimes governed throughout the sentence. Copperud opines that, strictly speaking, am should follow I, but says that "there is a strong tendency to use is, since am sounds artificial." Copperud's remarks confirm Strang's comment that this conflict is not yet resolved. (Webster's Dictionary of English Usage, p. 565)
Both is and am are accepted here.
Hi @Ash-Fred and thanks for the citation. It doesn't really apply here since Strang argues that the 3rd person pronoun, It, is what licenses the use of 3rd person verbs in all clauses. Yet, there is no 3rd person pronoun in the sentence "I, who is flying, am a man." In fact, following Strang's reasoning, not to mention basic subject-verb agreement, a 1st person pronoun in the subject position should license 1st person verbs in all clauses using that subject as a referent, resulting in the sentence, "I, who am flying, am a man." Approaching this from a theoretical syntax, descriptivist perspective, my native-speaker judgment is that "?I, who is flying, am a man" fails the test of naturalness and is, therefore, ungrammatical. I prefer ? over * to indicate ungrammaticality so as to not speak for all native speakers, though, I don't seem to be alone in this judgment. A quick search in COCA returned several occurrences of "I who am" and 0 occurrences of "I who is," where "I" is the subject of the main clause. Nevertheless, I'm sure Duolingo learners will appreciate your leniency.
Strang was probably only talking about the impersonal pronoun as we simply don't say "You are the one who are beautiful." I was rather just pointing out that, practically, subjects and verbs don't always agree with one another. Nevertheless, I have to stand corrected here; I asked seven people and five agreed with you, only one thought is was better, and the other one couldn't choose one.
"The truth is not a democracy." Be that all as it may, I'm still sure it is not incorrect to say "I, who is flying, am a man." In fact I'd say it all depends on the situation. If you're pointing at a photograph of you as someone who flies (날아라! 호빵맨?) you could very well say this and be correct to do so. Should they be so enlightened to such, I'm also sure native speakers would agree with me there. What grammar snobs and grammar deniers both so often overlook is the simple fact that grammar reflects meaning . . .