This is going to be a matter of opinion. I think that several arguments can be made for different points where Brian most resembles into a mature "man." For me, I would say that the development of "tough hope" towards the end of the book is where I think Brian passes through a portal of maturation whereby he becomes a form of "nature man." It is at this point where he no longer pines about not being rescued. He holds out hope for his rescue, but also represents a new found hope, something that is far more mature than anything else:
[Brian had] hope in the fact that he could learn and survive and take care of himself.
I think that it is at this point where Brian becomes a man because it shows that he is willing to accept the consequences of what has been dealt to him. At the same time, this moment shows Brian able to accept where he is and seek to survive. The prior chapters showed Brian immersed in self pity, and a sense of sadness about his condition. Yet, it is at this point where there is a steely resolve, something that reflects maturation in how Brian is able to accept what is there and endure, with an eventual triumph emerging from such perseverance.