This can get confusing. "Milan" is just one nation. I have 60 influence with Milan and recieve some culture. Milan conquers Panama City (a food CS). I still have 60 influence with just Milan. Panama City the player does not exist, and no one can receive the bonus food. For most benefits, it doesn't matter if a CS controls 1 city or 20, they give you same yields. There are exceptions, like a statecraft policy. The CS would also have a bigger army with more cities.