No civ can vote for itself, in order to require civs to vote for one another and so make cultivating good relations with other civs relevant to diplo victory.
What you do get with the UN, other than a countdown to victory if you've laid the groundwork, is get one free vote in your favour in addition to the votes you get from civs and city-states - this is why you see the UN icon with a 1 by it next to the player who's built the Wonder.