The various groups are just whitelists for the match making server.
Meaning when connecting to Mobius America you can only see those who also connects through that whitelist.
The idea for factions I believe, is that would allow for members on both sides of the planet to connect for the purposes of BGS work.
But all things BGS related are counted regards what platform or server you are using.
So what BGS numbers we see here, are the same as those in Open, as well as for those on PS4 and xbox, and all actions count against that whole inter-server whole.