I found P&W through TV Tropes so just for fun I decided to try to figure out what type of organization the top 10 alliances are. Hopefully you guys will find some enjoyment from this and maybe even look more into TV Tropes. Be careful! TV Tropes will ruin your life!
The original list
Rose is The Republic. Powerful and democratic it definitely qualify for the Republic.
The Syndicate, despite its name, is not The Syndicate (I hope) but rather a Mega-Corp. Not sure what to say... kind of self-explanatory... money good.
The Knights Radiants are the Badass Army. TV Tropes lists the Knights Radiant in the Stormlight Archives as a badass army as well.
The Immortals are The Alliance. Not exactly a rebellion per se but Tyrion did use to be in Fark (In all honesty I can’t really decide what they are. Suggestions would be appreciated and fun).
The Fighting Pacifists are Martial Pacifists. They may like peace but they can hit hard.
The Commonwealth is The Federation. Democratic with low taxes.
Camelot is the Vestigial Empire. Many people still see them as too close to IQ for comfort.
Grumpy Old Bastards are the Elite Army. I’m sorry I forgot you guys, please, don’t hurt me.
Children of the Light are The Church. They lean very heavily into this in their ranks and all that jazz.
Guardian is the Benevolent Precursor. Very old and decently respected alliance.
Welp, there we have it. If anybody disagrees or has ideas about what non top 10 alliances are let me know and I might add it to the list.
Mayor's suggestions
Oblivion is the Oddly Small Organization. It is a small, rich and very experienced group of ex-raiders from many well-known alliances involved in most wars.
Arrgh is the Thieves' Guild. Pirates, raiders, thieves and criminals.
United Purple Nations/The Covenant is/are the The Remnant. Ancient and fallen superpower.
New Pacific Order would have been the Hegemonic Empire, of course through IQ
Black Knights would also be a Vestigial Empire, they once were a pillar of IQ and a very powerful alliance with a lot of weight. These days they are more isolationist and a protectorate of the Syndicate.
Rose could also qualify as a Rising Empire since they have had many periods of strength and decline and one could say at this current moment they fit this trope. (also Resurgent Empire)
Pantheon would be in phase four of the Cycle of Empires
JadenStar10's suggestions
The Immortals is The Rising Empire. 4 members to 80 members in a month with 200 total members.
New Pacific Order is The Missing Faction.
Thalmor's suggestions
Knights Templar are Always Chaotic Evil.
Polaris is a Redshirt Army
BrythonLexi's suggestion
Advanced Syndicalist Mechanics is La Résistance. Lefties that eat whales.