Overwatch - So do we know why we're fighting each other again?

That's actually a pretty interesting concept. It would explain why we can have battles where the same hero fights themself on the other team, or how (in No Limits) you can have an entire team of just one character.
All the battles that take place in-game are literally simulated war-games by the Athena AI, millions of small variations on the same scenarios to hone her understanding of their capabilities and skills.

The real fights are in the cinematics, which are actual events that have taken place in the universe of Overwatch.

Aside from the need for some form of relaxing single-player mode, (where you don't have to worry about double-sniper/Bastion/Torb offense comps ruining your hardwork and/or feeling bad because your decision to go Symmetra on attack might have costed your team the game) this is the biggest reason for having a story-driven campaign in Overwatch. It's way too easy for someone who doesn't read the comics, let alone watch the animated shorts, to not know shit about who people actually are and what their relations to each other is, which makes all the cool little pre-game dialogue largely wasted. It'd also be a great way to practice certain characters if your only option is to go as them and your only task is to do what that character is supposed to do without worrying about the distractions of all the chaos involved when 12 people get together and start firing heavy ordinance at each other.