Curiosity is the mother of invention my friend...
Some more testing has revealed that the regiment is linked to how the spawnpoints (called "birth place" internally) are configured. If I set up a spawnpoint and assign it to the "Fat Cat Squadron" the "Regiment.name()" becomes "BoB_RAF_F_FatCat_Early" but if I spawn in from a spawn point I've configured for the "64th Sqn" the Regiment name is "BoB_RAF_F_64Sqn_Early". Then, if I don't configure the spawn point for a squadron (leave it at "<Any>") the Regiment name is set to "BoB_RAF_F_FatCat_Early". Seems the "Fat Cats" squadron is the default in this sim.
All this tells me that spawnpoints make up groups, not what squadron we belong to. Now, I know very little on how to build missions so please step in and tell me if there is a way to configure them in other ways. For example, I have no idea how to use the last tab - "Aircrafts" - in the spawnpoint property window.
In all my testing the callsign was always "Callsign0" but we'll see how all this looks when more players spawn in.
A related question for Kodiak (or anyone who cares to enlighten us): When I selected to create 3 airplanes the other two became AI's. I assume this is how to create groups, right? But how can players spawn in to that group if the server spawns AI's into those slots?