The first thing to note is that Rikki-Tikki-Tavi is a mongoose rather than a person (it should be noted that a majority of the characters in this story are animals: consider the two tailor-birds, Darzee and his wife, or the villains of the story, a married pair of cobras). As one can easily infer from the existence of marriage among the animals, these animals have been anthropomorphized so that their personalities are more in line with human beings than they are with wild animals.
Rikki-Tikki-Tavi is a young mongoose whose personality is best defined in terms of his bravery. This bravery is established both in his conduct around humans as well as in his fights against the snakes. He isn't fearless (for example, consider the scene where he kills Nag: he's very much aware in this moment that one wrong move can leave him dead), but he's described and characterized as being a fighter. He is very much a hero to the other animals and a protector to the human family he falls in with.