How does “How to Build a Fire“ display gender and identity throughout the story?

If I understand your question, the answer, regarding "To Build a Fire," is found in the traditional notions of manhood and male "self-sufficiency" London explores. There is also the theme of man and animal—their similarities, their bonding in a state of nature, and their differences or opposition to one another.
London himself, as his fiction in general indicates, believed in those traditional ideas about "masculine" values, but he also was critical of them. The Man in "To Build a Fire" tries to be too self-sufficient, traveling alone in the Yukon under conditions the "Old Timers" have warned him against. The ability or willingness to exist in the "wild," to defy nature, is not an unlimited virtue, in spite of what some men may believe. The Man understands things only on a superficial level, and he does not recognize his own vulnerability. Here is where animals, in their identity as beings relying chiefly on instinct, are in some way superior. The dog, with his ancestral knowledge, has a better understanding of the potentially fatal conditions at fifty below zero than the man has. Though the dog is fearful and obedient, when the Man tries to kill him, the dog's instinct overrides the normal dynamic between humans and animals. The dog survives—we are left to understand as he trots away that he will reach the camp safely—while the Man freezes to death, a victim of his over-confident male (and human) hubris.