Adventure
Kipling enjoys himself some adventure stories. His tales often center on action-oriented travelers going to foreign lands and risking all for the thrill and reward of dangerous circumstances. And Rikki-tikki's got all the adventure of an Indiana Jones flick, just on a smaller, more domestic scale.
The garden might not seem like an untamed wilderness from a human's perspective, but when you shrink your view down to a mongoose's size, suddenly the very same garden changes into a jungle. The mundane backyard becomes a land fraught with dangers such as poisonous snakes and summer storms that act like floods. And our protagonist mongoose must brave them all if he's to keep the British family safe and win the prize of a comfortable life. Let the adventure begin!
And we also have to mention another possible allusion under this category: looking at the bones of the story, we've got an intrepid hero who kills a scary (male) monster and then follows a second scary (female) monster back to her lair to kill her. Hmmmm. Sounds to us a lot like Beowulf—the prototypical English adventure story, if there ever was one.
Coming-of-Age
"Rikki-Tikki-Tavi" follows the typical coming-of-age model for a boy's adventure story. Well, typical in that a boy goes from being a boy to being a man. The fact that the protagonist is a mongoose is a little atypical, but, hey, mongooses have to grow up too.
Like a typical boy's coming-of-age story, Rikki-tikki starts off a young mongoose who has never left his parents' side. Thanks to a summer storm, he's whisked away from them and enters the care of a British family. In their garden, Rikki-tikki takes on the role of adult mongoose and does battle with the resident snakes.
At first, he makes some mistakes such as not killing Nagaina when presented with the opportunity (30) or thinking Karait less dangerous than the cobra (35). But as the story continues, the mongoose grows in skill, courage, and confidence. By the end, he's a snake-chomping powerhouse.
Fable
Kipling's definitely tapping into some Aesop with "Rikki-tikki-tavi." Just like a classic Aesop fable, the animal characters teach a moral meant for the human readers to follow. In the case of this short story, we're taught that humans should take on the characteristics of Rikki-tikki—courageous, pragmatic, adventurous, curious, action-oriented and protective.
Of course, what's a fable without a foil: an ant to compare to the grasshopper, or a hare to a tortoise? The other animals exist to provide a counter example to Rikki-tikki—as in, don't be these guys.
Darzee is a flighty bird who is more interested in creating songs than doing useful things to protect the garden. Yeah, don't be that guy. Chuchundra is a muskrat who slinks about at night because he's afraid of Nag. Don't be that guy either. Instead, be Rikki-tikki and be awesome.