It varies from shooter to shooter, of course. Just like for any other genre. But let's be clear that Smash and Splatoon really have very little in common, so it wouldn't make sense to pull anything from Smash.
For other shooters, here's how it works: tournaments establish a number of "gametypes" that can be played during a match. A gametype is a combination of a multiplayer mode and a map. So, for example, Turf War on Saltspray Rig would be considered one gametype.
Then, the organizers essentially plan out which gametypes are played for which game and during which round. So, for example, a gametype list may look a little like this example:
Round 2 (Winners Bracket)
Game 1: Turf War/Saltspray Rig
Game 2: Tower Control/Urchin Underpass
Game 3: Splat Zones/Walleye Warehouse
In this example, two teams are playing a best of 3. These are the gametypes for each game. There's no counterpicking or stage striking like in Smash - everything is laid out beforehand.
Everything else is pretty simple - certain amount of players, they pick their sets, they play their matches using the gametype's win conditions.