Okay, so I finished reading the Factol's Manifesto, and I have determined that the answer is that the material is inconsistent.
The Factol's Manifesto gives actual numbers for some of the factions, and describes initiation and the duties of new members. According to the way it presents it, every Harmonium member really does go out on guard duty. And their factol's 14 year old daughter wants to become a member when she grows up.
So the premise that it presents is that most citizens of Sigil aren't members of factions.
But then you have the other books which list just about every NPC as a faction member, plus you have the rule in the original boxed set that all planar PCs must start as faction members (of course, that could just be the 1990's era RPG writer domineering), plus you have this text about the Free League in the original set: "There's nobody who's got a sure key to the truth, so it pays to keep the options open. Maybe the multiverse is like the Lost say, but it could be the way the Godsmen tell it. Side with one view and find out it's wrong and, well, a fellow comes up a loser. There's no wisdom in that! Still, a body's got to belong to something, if he wants to stay alive."
Now, that is presented semi-in-character, but it doesn't change the point. The original boxed set implies pretty much everyone is in a faction, and the NPCs throughout the product line back that up.
In this case, I think I'm going to call foul on the Factol's Manifesto. It contradicts the greater weight of material.
It actually seems to enjoy contradicting material. As an example, comparing it to In the Cage, we find a different setup for the City Court (among other things). In the Cage has non-Guvner judges, while Factol's Manifesto has a more tightly controlled Guvner only legal system.
What is annoying is that these products literally came out a month apart, and they can't even get their story straight.
While I'm at it, the Factol's Manifesto seems to take a messed up view of the Indeps. The original set implies to me that they are pretty much the faction for people who don't want to commit to a real faction. It describes them in basically faction agnostic terms; and even says some of them have their own distinctive views and might want to start their own factions. The Factol's Manifesto, on the other hand, describes them as these philosophical individualists with an extremist level of zeal. I think they really missed the original intent on that one--but I suppose they needed to do something like that if they were going to make factions be a minority.
If I hadn't had as long to thing about it and read the discrepancies, I probably would have went with factions as a minority, just so that they aren't permeating everything. But after thinking through what actually makes sense for people living in the setting, I think I'm going to go with the factions = religions premise we discussed, and just pick and choose which material I like from the books.
Apparently, the discrepancies in the Factol's Manifesto are a "known issue", or at least this is what Monte Cook (author of the original set) had to say about it:
"As for Factol's Manifesto, some of the sections I think are great expansions on the faction background, but there are a few (particularly the Mercykillers, unless my memory is toast) that seemed to wildly miss the point, even to the extent of making the faction unplayable."
That really got me to want to carefully review the original presentation of each faction after reading their Factol's Manifesto entry to try pin down which ones might have missed the mark.
And a random question, in case anyone happens to know the answer: What is the point of the City Armory?
Seriously, the books all agree that the Doomguard keep the weapons away from the Harmonium...which are basically the people who the Armory should be supporting. Who else would an official city armory be supplying but the authorized law enforcement or military for that city? The Factol's Manifesto has them basically just selling them like a giant weapon's shop, which seems absurd.