In Justice League: War, why did Superman and Batman fight and how did Green Lantern know Batman? Was this some kind of alternate universe or something?
Most movies and cinematic retelling of stories may change what is currently established by the more recent or relevant timelines. In Justice League: War, the story is a retelling of the New52 Justice League's first meeting with Darkseid.
- The comic pretty much acts as a storyboard for the confrontation: Here are two panels indicating the scene just before Superman wipes the floor with Green Lantern and Batman. One is from Justice League #1 and the second is from Justice League: War.
This is the setup right before Batman and Superman fight in Justice League #2.
The story is a retelling of Justice League #1 and #2 (2011):
In this story, Batman, Superman and Green Lantern don't really know each other from first hand experience, only what they have heard on modern media. Batman tends to know about EVERYONE before they know about him. Green Lantern didn't know Batman, he only knew OF Batman, primarily as an urban legend.
The first confrontation between Superman and Batman takes place because Superman is an alien and Batman suspects the invasion that is taking place is also an alien event. It takes a minute before he realizes Superman is not involved in it.
Superman, on the other hand is not sure what is happening and is trying to figure out which side everyone is on. Batman and Green Lantern's unexpected visitation to Metropolis starts them off on his bad side because he just discovered the aliens as well.
This is the comic trope: Mistaken Identity - Superheroes fight before teaming up. A standard method for introducing new characters to each other and to new readers. Since this story was directly after a major DC reboot, none of the characters are familiar with each other, so it gives us a chance to see how powerful they are and what their capabilities might look like in this new DC Universe.