You can't "register" a faction. The game knows about the factions it knows about and is not interested in anything else.
Are you meaning you want to add some random not in-game identity to the reputation frame and generate your own reputation based on "stuff"? Although it looks like you've done that with the "my sub-faction" in the image.
Possibly you just want to update your personal reputation status bar etc when the reputation frame is displayed which you could do by hooking the ReputationFrame_OnShow function.
Lua Code:
hooksecurefunc("ReputationFrame_OnShow", function(self)
-- update your widgets with the latest from information you've presumably stored in a table.
end)
You could also use
Lua Code:
ReputationFrame:HookScript("OnShow",function(self)
-- update your widgets with the latest from information you've presumably stored in a table.
end)
Which is pretty much the same thing but works if the script doesn't call a separate function.