The central conflict in Three Houses is, at its core, a conflict between Law and Chaos . Rhea/The Church represent Law and Edelgard represents Chaos.
Honestly, it seems like the writers straight up used Shin Megami Tensei alignment logic, just look at the descriptions from this page
Law: This alignment is associated with God, order and peace. In the game, it is represented by the Messian religion and the colors blue and white. The Law alignment promotes order and safety, but taken to an extreme, it leads to dictatorship and elitism.
Chaos: This alignment is associated with Lucifer, freedom and war. In the game it is represented by the Gaean religion and the colors red and black. This alignment promotes freedom of choice, thought, and action above all else in stark contrast to the controlling nature of the Law alignment.
So, let's look at the stuff I've bolded:
God, order and peace: Pretty self-explanatory as to how this represents the Church.
blue and white: Church. Rhea. Blue Lions. Faerghus.
dictatorship and elitism: Edelgard fights to end Rhea's tyrrany and the elitist Crest system
freedom and war: Self-explanatory. Edelgard starts the war, with one of its main goals to free humanity from Rhea's regime.
red and black: Edelgard. Black Eagles. Adrestian Empire.
Lucifer: There's your reason for the horns. Symbolism.
As an added bonus, let's look at the description for the Neutral alignment, again from the alignment page linked above.
Neutral: Also known as "Balance", this alignment accepts the need for laws to govern the world but not at the cost of personal freedom. It rejects the concept of relying on Lawful or Chaotic powers to focus instead on personal empowerment, refinement, individuality and the inherent strength of mankind. Although presumed to be the most idealistic alignment by many, it is also the hardest alignment to attain because of its isolationist nature, destroying everything in your way and even standing against former friends.
Read that description and consider what happens in Crimson Flower. It's why I consider it the "Neutral" ending of the game.