What is the meaning of the penguin in Fight Club?

The cave and the penguin are the result of therapy where the female voice tells the whole group how to "meditate". Every person has their own interpretation of the cave and an animal that is inside (which represents them); in the narrator's case he sees himself as a penguin.

He probably interpreted himself as a penguin to show himself his inability to let go of his depressing lifestyle, since penguins can't fly away from their "problems" they can merely slide away. Which is what he did he slid to the "new life".

In the movie there is also another quote that references the "sliding" :

Narrator: You had to give it to him: he had a plan. And it started to make sense, in a Tyler sort of way. No fear. No distractions. The ability to let that which does not matter truly slide.

The penguin was the Narrator's "power animal." As Arremer, it told the Narrator to slide to mean "relax and let go." Further, I think the choice of a penguin was based on it being black-and-white to hint at the duality building within the main character.

"slid could also represent how he slid into a new life each time he went to a different class like how he is Jack outside of the groups then Cornelius in one support group

According to Native American tradition if you have a Pinguin power animal: "You have the ability to create anything and everything you choose in life. You understand the concept of teamwork and utilize it constantly in order to get what you want. You know exactly where you are going and what you are trying to accomplish at all times. You move easily between the physical and spiritual realms and often have a gift for Lucid Dreaming. You are polite, proper and never forget your manners. You enjoy cultural exchanges and high societal affair. Many of you are diplomats and politicians" It's everything Tyler becomes.

Penguin says slide as the solution to the narrator's miserable emptiness at chasing the chains of modern existence.

Marla, the nemesis intrudes all of his favorite group therapy hang outs becomes the inhabitant of his internal safe place then carries the same line: Slide.

Let go of the struggle (hand burn and car crash scene) stop trying to conform because it is destroying you draining you of identity, mind, and dreams, finally purpose.

Tyler who at least pre-existed Marla Singer "that sore in the roof of your mouth that could just heal if you could stop tonguing it" is the embodiment of the word slide, the solution for the misery of the narrator. He is the protagonist natural man to the antagonist unnatural corporate man who has no purpose but to serve his chains, his master, his destroyer of his spiritual state.

The only means of casting off the materialist slave state is to let the credit cards, the apartment, its contents, the job, the illusion of polite society slide. It is all insane anyway. It has no moral justification no moral money system of actual set value, let it all go and revert back to the natural state that everyone is resisting so hard, and claims some false moral superiority over.

This is why I go by Slide as my street name.