It's not an official feature in the game, but a few expansions have introduced new worlds with a premade sim as the mayor, so in my head, almost every world has one now - usually, but not always a played sim in the politician career who somewhat represents the world itself. For example, my Willow Creek is dominated by one big family in particular, so of course someone from that family is the mayor. Oasis Springs is the place where humans first made contact with aliens (I've edited the world itself and its description to reflect this), so the mayor is an alien. The only worlds without mayors are San Myshuno, because I haven't thought of one yet, and Magnolia Promenade, which I view as a district of Newcrest.
I think it would be cool if we had a trait or something like that to appoint one of our sims as the mayor. But of course, there can only be one per world, so if you assign the trait to a new sim in a city that already has a mayor, the previous mayor loses it. What are your thoughts? Do your cities have mayors?