It seems like you only get the culture type of the previous owner of a city, not the original city builder. If that was the barbs you get nothing. The barbs themselves were theoretically getting the ability to build the culture specific things of whoever they took it from - but that is theoretical in as much as I think they are blocked from building any of the relevant buildings.
It might be doing this on purpose. Or it might be a bug. Perhaps barbs are supposed to wipe out the culture of any city they capture. Perhaps they are not.
I'm not sure what happens if the intermediate owner was not the barbarians since in C2C I don't remember taking a city from someone who was not the original owner other than barbs any time recently. It might not be an issue only with them, maybe you always get access to the last owner's culture? Or it could be barb specific. (As a bug source possibility, perhaps when it does the check for barbarians, since they have no native culture and therefore may have special code to deal with them, it is checking the barbarian status of the previous owner instead of the original owner which leads to it setting the uniques for the city to you instead of leaving it as the original non-barb owner's set. Or something like that.)