Look at the very last sentence of Damon's email, that's what I'm basing this on:
"Neutrals, as the rule you quoted says, do not belong to any faction. That solves all the questions you asked regarding neutral cards. You cannot compare something that does not exist to something else. Neutral cards have no faction so they have nothing to compare, in the case of Communal Shower that means you can commit a Cthulhu character and a Neutral character both to the same story, no problem. A Neutral card does not belong to the same faction as another Neutral card, because they belong to no faction at all.
Strange Delusions will give a character an affiliation of your choice but it does not say anything about removing existing affiliations, just adding one. So a card that was Cthulhu, could be Cthulhu Hastur. It would match both factions and treated at all times as if it were both factions until Strange Delusions effect ended with the phase it was triggered. If a card said Cthulhu cards cannot commit to a story, that card could not commit. If a card said non-Hastur cards do not count their skill, it would count its skill because it is a Hastur card. Communal Shower would prevent it from committing to any story at all since it is a character from different factions."
The character counts as a different faction from himself according to this. Logically, I believe that means he is also a different faction from other characters that share his same faction(s). If you want to put it into more precise terms, I would have to say based on Damon's reply that two characters are "the same faction" if and only if every faction of character A matches every faction of character B.