Mutual recognition, as a Hegelian term, has been misunderstood greatly. It is not something great, it is not static, but it is precisely in mutual recognition that differences emerge.
Fight Club revolves around recognition. The first scene with Bob and the narrator hugging together exposes already the hidden inauthenticity in recognition—the two parties are recognizing each other only partly, never whole. Both are hiding something, and can never reveal themselves fully. (Bob, at the end, decided to leave the support group because of this.)
Then is the actual mutual recognition between the narrator and Marla. This is the recognition of what cannot structurally be recognized in the support groups: the hidden obscenity that lies outside of the Big Other’s gaze. (Which is also why Chloe, who was soon going to die, was kicked off the podium as soon as she started talking about her real desire (for sex)—desire unmediated by the Big Other.) In this recognition the narrator understands some part of himself that he has been fetishistically disavowing (I know very well… but). This recognition is a moment of violence—the narrator cannot get off anymore after losing this distance with himself.
It is in this recognition with Marla that Tyler—defined against the narrator whilst also implicated with him—develops. And this leads to the last recognition at the end of the film, where the narrator and Tyler recognizes each other as the same. But it is this recognition that blows them apart with the narrator shooting himself in order to shoot Tyler.
Each recognition leads only to difference. Identity, the a=a, already implies difference in the a splitting into two sides. There is no real mutual recognition that is the end of history. There is only the difference out of the a=a.