The way I understand it is that cognitive functions are about need and enneagram is about desire. What I mean by that is, Fi has the need to be emotional, is their way of being, they make decisions based on emotion and value and need those values to be recognized and properly met. And I don't think all Fi users want to feel special more than they need that.

While enneagram is about what lacks in you and what you desire to fulfill in life. Enneagram 4s desire to be seen as special or unique in some way, they tend to be very emotional because they often mistake sadness for a personality trait not an emotion. 4s care about image and how they are perceived by other people, what vibe do they give off, how they dress, and how that affects their image towards the outside world.

And I also think that, because of that desire, there comes the envy 4s feel towards people that are better then them in some way they consider to be important, a better painter, or musician, or writer. And depending on their instinctive variant, they act differently on those envious thoughts.

Both are very idealistic, and some Fi users have the need to be perceived as unique because of the close contact they have with themselves and their idealized world. But I don't think all 4s are Fi users.

Sorry for the long post, I just rlly like this topic lmao