Crew battles are indeed a thing, but typically small tournaments such as single day locals don't have them, though you might see like college vs college. They are more likely the bigger the tournament and are mostly seen in large 3-day tournaments. The teams typically consist of the top ranked players and not volunteers. The parameters can vary as there things like character crew battles where a team consists of the top players for a particular character and everyone must play the same character on the same team (i.e. Fox's vs Sheik's). Most of the time it's region though. Continent vs continent, country vs country, state vs state, etc. It usually depends on the player base as a tournament in California can likely have a North CA and a South CA team as many players from that area would be in attendance. So you can see things like NorCal vs SoCal vs East Coast vs Japan. The region they choose is heavily dependent on quality and number of players from a given area in attendance at a given tournament. There isn't anything concrete about how teams are formed so it does vary a little, they can be seemingly random as you could have a Team Pepsi vs Team Coca-Cola crew battle based on which soda you prefer. But more times than not it's region.
As a friendly fyi crew battles are also most often played on a Friday of 3-day event, usually as a way to wrap up the day. It's something that kinda adds excitement to the first day of a tournament which is typically a less exciting day for viewers. Things like a tournament ending in a crew battle such as Civil War ZeRo vs Ally are rare.