You can work for who you want, when you want. It's nice to coordinate if they are actually around.
The implicit corollary to that is that any PMF can have allegience pledged by any number of Squadrons... this is just one little factoid demonstrating why pledging allegience ultimately means nothing in the context of ownership or representation of that
group faction.
I had not heard that the BGS was shared between platforms, that is good to know. I've sent out a FR to a Commander on Inara that has the same name on PC, but no reply yet.
A follow-on from that is why people who suggest OO-BGS (or that you can only affect the BGS when in Open) under the claim that players should be able to stop other players affecting them are off the mark.
BGS is that living, breathing universe that moves when players touch it, regardless of mode, platform, or location. Even though XBOX/PSN/PC players can never interact cross-platform, they share the same BGS and the effects that occur. The fact player groups compete over it is a secondary effect.
Think of it like the gambling industry running bets on who is going to win an Election. An Election is a process which a government can use to elect it's new leader... but people gamble on it. That's fine, but it would be ridiculous for the gambling industry to lobby a government to change it's Electoral process to better facilitate gambling. It simply wouldn't make sense.
Sorry, bit of a tangent.
EDIT: Struck group and replaced with faction, because for clarity:
Group: A collection of players identified by FD as "a group", represented to FD by their leader whose email is on a distro
Faction: An NPC collective body of citizens ascribing to a particular government type.
Squadron: A collection of players (Pilot's Federation Commanders) who have banded together for some collective purpose.