There are definitely commonalities, but there's nothing "natural" about this election, or any part of this process. Honnouji Academy is Satuski's world, and everything within it revolves around her and furthering her goals. People aren't fighting for survival; they're fighting to advance, to get closer to that shining light of the center, the guarantee of power and strength. Satsuki is encouraging this process to create a stronger organization for herself, and for no other reason.
Nobody has to do anything for natural selection to happen. It's the product of an uncaring and impartial environment, altered by factors outside of anyone's control. In contrast, the competition of modern society, and the competition in Satsuki's academy, are artificial forces harnessed by the powerful to ensure more efficient production or strength. Complacency and peace lead to weakness. If everyone below the highest rung is fighting each other to serve the whims of the leader, they must always be strong enough to defend themselves, and they have no chance to aim at the one on top. To call back to the show's heavy-handed references to the Nazi Party in the first episode, this is how many totalitarian dictators operated.
Essentially, natural selection and Social Darwinism are both passive theories. They justify the world as it is. They're fundamentally conservative ideas. Satsuki is actively sowing chaos for her own ends.
It's perfectly ordinary for proponents of Social Darwinism to 1) exert great effort to restore or recreate the state of nature among what they perceive as corrupt society and 2) to display a certain amount of hypocrisy when it comes to their own interests. Bioshock is merely the first example that comes to mind.