My cousin just got his first passport and I'm trying to convince him to get a new one as I'm sure the "country" part on the info page of his passport is wrong or maybe I'm wrong. So on the country born line it says: Dubai/ ARE...in his application he put "United Arab Emirates" so shouldn't his passport say U.A.E. or is the country code "ARE" for United Arab Emirates for a Canadian passport?
ARE is indeed the ISO 3166-1 alpha-3 country code for United Arab Emirates. These are the codes used for machine readable passports as defined by ICAO. Looking at the wikipedia page showing the format used for machine readable passports you can see that the ISO 3166-1 codes are used both to identify the issuing country as well as the nationality of the passport holder. So your cousin's passport seems to be correct.
For completeness sake note that ARE is also the NATO 3-letter code for United Arab Emirates.