It's a strange thing about English. "Kurd" is a noun in English (and needs an article). "Kurdish" is the adjective (which doesn't need an article). So you can say "I am Kurdish," but you have to say "I am a Kurd."
There are several nationalities that work similarly (Turk/Turkish, Swede/Swedish, Finn/Finnish).
There are others with different patterns: "I am French," but "I am a Frenchman/Frenchwoman/French person"; "I am English" but "I am an Englishman/Englishwoman/English person"; "I am Spanish," but "I am a Spaniard."
There are others where the noun and the adjective are the same: you can say "I am an American" or "I am American," and the same with Canadian, German, Italian, Mexican....
It's a confusing, and basically has to be memorized for each nationality.
**EDIT: As I think more about this, I think all nationalities with "-an" endings (Norwegian, Ecuadoran, etc.) can function as both nouns and adjectives, and all nationalities with "-ish" endings (Irish, Turkish) can only be used as adjectives. That should help a little, although it doesn't tell you what the noun form is for the -ish endings.
Why is "I do not speak Kurdish, but I am Kurdish" wrong?
It is not wrong but back to front to Duo's Turkish question.
"I do not speak Kurdish but I am Kurdish" - Kürtçe bilmiyorum ama Kürt'üm. - I understand this & it is grammatically accurate.
I am persian and I should say Kurdish people fall into two categories. A bunch of them live in West Iran and they are so kind and hospitable. bunch of them live in iraq that I actually dont know about them but I think they are very good people
ural-altaic language family theory has been rejected by most linguists for a long time, although it is still taught in Turkey and many people believe in it. Although they are structurally and grammatically similar, they have almost no common vocabulary (of course there are a few cognate words)