They join one airline, but they can fly any of the flights for any airlines which are listed. That just serves as their "home" airline. An admin can change that home airline. It's really just for their pilot ID. I think setting up separate installs is necessary, you can just use the multiple airlines facility.
You can do schedules specific to each airline within one install as well.
For registrations, you'll have to add something (like a -C for classic, or something) to denote which registration the airline belongs to. The reason you can't do multiple is because of ACARS applications, I can't differentiate, but adding that identifier will help.