Rikki-tikki starts the story as a young and inexperienced mongoose, which this means he has only one place to go from there: up. Yes, like many coming of age stories, "Rikki-Tikki-Tavi" is about a boy who grows into manhood by having himself an adventure. The adventure tests him every step of the way, and by overcoming the trials, the boy mongoose learns about himself, develops his skills, and discovers his reason for living to become a man mongoose. Sure, Rikki-tikki is a little furrier than our usual coming-of-age hero, and his journey only takes him as far as the backyard. But trust us. There's some major maturation going on.
Coming of age is the major theme tying "Rikki-Tikki-Tavi" to the rest of the Jungle Books stories.
Rikki-tikki doesn't actually come of age in the story. As the narrator points out, he's a "true mongoose" (7). This means he's not maturing so much as doing what a mongoose does best, snake killing.