This is interesting. I struggle with "is Arab" too, but from a bit of searching around it seems fairly consistent that "Arabic" relates to language and culture (only), "Arabian" refers to the peninsula specifically (ie an Egyptian would not be Arabian), and "Arab" is an adjective for anyone who speaks Arabic as a mother tongue. I had not seen Arab used as an adjective equivalent to say "Spanish", but it seems that is exactly what it is, so perhaps Zayd is Arab is entirely correct...
"Arab is an adjective for everyone who speaks Arabic as a mother tongue"
No definetly not. A lot of people would be really offended by this. You just stamp their entire culture by calling them Arab just because their mother tongue is Arabic.
An example is Assyrians, some Assryians do speak solely Arabic but that doesn't make then Arabic.
In my opinion, it would be like saying "Zayd is Spanish" vs "Zayd is a Spaniard". Both have a similar meaning, but use different words. At this point, I believe that Duolingo is trying to emphasize learning and using the correct WORDS, rather than using acceptable meanings with different words.
I find Arab unusual used as an adjective in a complement*. If you said "The student is Arab" that would sound wrong, but "...the Arab student" would be ok. If you wanted to use it as a complement, it would be more usual to use the word as a noun, "The student is an Arab".
I know that Arabic is meant to relate to the culture and language, but I think it is probably used for people more than the others, in practice.
I have never heard Arabian used in normal speech for describing people. I imagine that the implication would be Saudi, rather than relating to any of the inhabitants of the peninsular. I am not sure that the Omanis and others I have met would use that adjective to describe themselves, if they were speaking English. And Saudi is a more normal one for an inhabitant of Saudi Arabia.
As for Arab, as Akkad890034 says, you have many races who speak Arabic. There are many Amazigh, (Berber) people in N Africa, who speak Arabic as a first language, but are not Arabs. It's a word that refers to a race of people.
*Complement is a grammar term- nothing to do with flattery. Basically, if you have a sentence with the verb "to be", you will often have a subject, then "to be" and then the complement- the bit at the end- maybe a noun, maybe an adjective, often a short phrase. In "I am a doctor", "He is irritating", "They are cabbages", the complements are "a doctor", "irritating" and "cabbages".