If you are dealing with philosophic issues concerning ethics, characters will have beliefs and behaviors related to those topics. Even if the beliefs are Moral Error theory. Depending on how much the play touches on that subject, avoiding using words for the subject will grow frustrating and tiresome. Eventually one would start using words to describe the beliefs, the behaviors, and how those beliefs / behaviors deviate from or emulate moral truth.
Notice this is not the language of a faction system, the usefulness of the tool is having a common language and multiple depths of summary so the right amount of detail can be used at the right time. In some cases I would describe O-chul as a moral individual. In other cases I would dive into more detail to differentiate them from Roy or Elan. Or I might go into even more detail about a particular choice O-chul made. It is not saying Elan and Roy are on the same side, it is comparing and contrasting their roles as moral agents.
So alignment does not presume evil people would be prone to work with each other. Nor does it presume _____ people are prone to work with each other.
I don't think it is as difficult as you lay out. So I recognize the underlying tendencies but I come to a less strict conclusion. Organizations or movements can form around a particular immoral behavior. Furthermore depending on the rationalization for their own actions, they might be desensitized to another type of action.
For example a self righteous mistaken individual might be out to purge all ___. Someone that likes general slaughter would be willing to work with them until the victims dwindled. Someone out to acquire wealth might tag along to seize the assets of the victims, they are going to die regardless so why not profit? Etc. Three individuals all willing to kill for completely different reasons. The greedy opportunity might even find the other two abhorrent, but their victims are doomed so the opportunist considers their blood soaked hands to be clean.
Evil will not team up merely for the sake of evil, but evil individuals understand the concept of cooperation (also exploitation, betrayal, and love).
I guess I should also mention that people vary in how morally tolerant they are. A common thug might want to work with some saints, but the saints might not want to work with the thug. In search of allies the thug might lower their standards. Segev went into more detail here, so feel free to only reply to their section.
I don't see alignment as a faction rule system which is part of why I asked, so I will ignore the final sentence.
I see, so with a decent social rule system framework, a faction system helps codify career mechanics in a way that helps set useful expectations between the players. Sort of like how an example of a skill usage with a DC could set useful expectations. Thanks for elaborating, that helps.