It isn't a trick. It means what it says. What if somebody said the doctor is a nurse. You could reply, no, the doctor is a doctor and the nurse is a nurse. I guess you haven't run into enough idiots in your life. Give it time. You will find there are people you have to say such obvious thing to. Although, if you haven't met such an idiot yet then it is very likely that...well, nevermind.
It's trying to teach the basic form of one of the two basic sentence types in Arabic, the nominal sentence.
Verbal sentences, any sentence with a verb, can be just a verb, or have nouns also, and they can come in any order, but traditionally VSO was considered standard. SVO is very common today.
But any Arabic sentence that lacks a true verb, like this one, which simply combines nouns and/or adjectives is called a nominal sentence, with the basic form "subject + description". The subject and the description can each either be a noun by itself, an adjective by itself (which can imply a noun), or a combination of nouns and adjectives.
It's also trying to show us case agreement between nouns (and adjectives, which work basically the same way). In standard Arabic (but NOT spoken, vernacular Arabic) nouns and any other adjectives describing them must all have the same final vowel, unless it's the last word of a phrase or sentence: "u" if it's the subject, "a" if it's the gramattical object, and "i" if it's in the genetive case. (For more on the genetive case, which shows posession and relation between nouns, google "iDafa")
Nouns and their adjectives must also agree in definiteness, so if "al", "the", comes before the noun, it should also come before the adjective, or alternatively, if it's indefinite, the noun and its adjective(s) should both end in "n", after the final vowel (only in standard Arabic, and again this is dropped off if it's the last word in a sentence or phrase).
BUT as you can see by this sentence, even though a subject in a nominal sentence must have the same final vowel as the noun(s) (/ adjective(s)) describing it, it does NOT have to agree in definiteness, (or number or gender). And again the final vowel mentioned before gets dropped at the end of a phrase. That last part is pretty crucial for showing where the end of the sentence is in standard Arabic.
In dialects, the Ns and vowels at the end of the word aren't normally pronounced, but Duo hasn't really figured out how to teach those differences yet.
It doesn't have to do with definiteness. The u sound at the end of al-mu3allim-u says it's the subject, but even if it were indefinite, it would still have the u sound if it were the subject. It would just be mu3allimun, a teacher, with the n at the end and no al- at the beginning. In fact the second word in the example would be pronounced like that (since in sentences of the form "[something] is [something]" the two things are always the same case), except the case marking vowel and the n are never pronounced at the end of a sentence or phrase, so in this case it's just mu3allim.
If al-mu3allim is the object of some action, it becomes al-mu3allim-a, and "of the teacher" would be al-mu3allim-i. But you could also have the indefinite, mu3alliman, if "a teacher" was the object of a verb, or mu3allimin if you were saying "of a teacher".
-hu is a completely different word. It's an object pronoun meaning "him" or "it" when it is attached to the end of a verb, and it's a posessive meaning "his" or "its" when it's attached to the end of a noun. It's not a copula. Copulas are words like "am" and "be" and "is", and Arabic doesn't really have those. The closest might be the verb root k-w-n which is more like "to become" than "to be". It's also important to realize -hu changes to -hi after the short i vowel, the long i vowel, and the ay diphthong. It's also often pronounced as -u, without the h sound, after a consonant.
And the last piece of the puzzle to make this all make sense is that the case marking vowel and the indefinite n are only really ever prnounced in formal Arabic, and never in casual speech. So if someone just says "al-mu3allim-u", it's actually unclear whether they're saying "the teacher" in formal arabic or "the teacher of his", "al-mu3allim-(u)-(h)u", in spoken vernacular form.
Why "the teacher is a teacher" is the solution and not my idea: the teacher is teaching?
Okay, I guess it's about noun vs. verb, different from even in Arabic (of course :D), but somehow this is what sounds logical to me...
I was lucky that the word teaching wasn't an option to chose from words, but otherwise I would have translated freely by myself like a verb...
Actually there are so many stupid meaning examples sentences here, but it's okay: it's the same in every language book too... :)