Has the feminist movement caused problems of gender identity for these men? Please explain.
I think that much of this answer would have to depend if one buys the initial premise of the work, in general. If one believes what Tyler/ the narrator suggest, then, yes, the assertion of women's roles into society causes a fundamental sense of questioning in men and causes a gender identity issue. However, I think that this might be a bit too reductive. It makes for interesting reading and helps to advance the plot with such an idea. Yet, I think that such an argument makes rights and the assertion of voice a "zero sum" game. In this, the advancement of women seems to trade off with men. I am not entirely certain that is accurate. The narrative of America has been filled with individuals and voices that have moved from margin to center. For example, it did not cause a massive crisis in voice or identity when poor white Americans found Andrew Jackson to be a champion for their cause. The transformation of the Presidency as one protecting and representing the elite and upper echelon to one that is more representative "of the people" did not cause a seismic chasm in identity. America has been a nation where more people are included and this process of understanding and re-acclimation is not one where loss of identity is so finalized. Rather, it is a transformative process. The individuals who belong to Fight Club or Project Mayhem do not do so because of the rise of feminism, but rather out of a sense of belonging. It happens to be that the feminist movement is a source that they perceive as frustrating, but I don't think that it is reasonable to presume that the results of the feminist movement has caused a change in identity.
However, perhaps, one of the lasting legacies of the work is precisely this point. If individuals believe it as much, there is a level of acceptable truth within it. The men who belong to Fight Club have not really ruminated and examined their own sense of consciousness. Rather, they are fine with Tyler/ the narrator doing it for them, and in this, there is a belief that the rise of women through feminism has helped to explain their own realities. It is here where I think that the men, faulty logic and all, might actually believe that feminism is the reason why they are how they are, as opposed to engaging in reflective thought. Reductionist thought ends up becoming the final legacy of these men, regardless of the presence of feminism.