"No, I am not" doesn't make sense in the way they have taught you up to now. "No, not me" is a correct phrase in English. You cannot interchange "No, I am not" and "No, not me" they mean two different things.
Not always true. "Are you doing that?" can be replied to with "No, not me" or "No, I am not" short for "No, it is not me doing that" or " No, I am not doing that"
Because the verb "to be" is not in the sentence.
No, I am not! - Nein, bin ich nicht!
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My native language is Portuguese, but I'm an English Teacher an I'm using Duolingo to improve my German. In Portuguese, we have an expression very common that is: "Não, eu não.", which means "No, I not" literally translating [not used in English], or "Não, não eu", translating "No, not me". This expression doesn't use a verb in Portuguese. The same thing happens in German. "Nein, Ich nicht." Since there is no use of a verb, the translation in English couldn't be "No, I don't" or "No, I'm not", because these expressions use a verb. The only expression with similar meaning that could be used in any case would be "No, not me."