If you are asking if "Am I girl or boy?" should be accepted, it should not. You would not, for example, say, "Is it hat or coat?", would you? You could say, "Am I male or female?" or "Is it red or blue?" -- but those are different. In those two sentences we have adjectives, and the article is not required. In the exercise sentence, we have nouns, and the article is required, though it can be omitted for the second noun. "Am I a girl or boy?" is also accepted.