You are correct that civilizations are not the same as "teams" in a lot of other games. They are a collection of players with an incentive to work together to beat other collections of players, but you are still competing against the players in your civilization!
To put it in terms used in other games like Civilization 1 through 5, you can consider a CivWorld civilization to be like a collection of players who are all in a non-permanent alliance. You're all working together against common foes, but in the back of everyone's minds is the fact that in the end, the players you're working with are still your opponents, and when the time is right, you should do what is necessary to win you the game.