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.
When they say X is Y, there is no verb connecting the two things. It's just the way Arabic grammar is. We do that sometimes in English, as in "My father is tall, my mother, short."