I think one key human-like trait of Rikki is the fact that he is insatiably curious. He's so curious that he isn't necessarily afraid of things that he should be wary of. Kipling tells his readers about Rikki's curiosity fairly early on in the story.
It is the hardest thing in the world to frighten a mongoose, because he is eaten up from nose to tail with curiosity. The motto of all the mongoose family is "Run and find out," and Rikki-tikki was a true mongoose.
Readers learn from Rikki's actions that he is very curious as well. In fact, Rikki's curiosity is what keeps him at the human house in the first place.
"There are more things to find out about in this house," he said to himself, "than all my family could find out in all their lives. I shall certainly stay and find out."
Another human characteristic of Rikki is his bravery. There are times when Rikki is fighting for his own personal protection, which demands some bravery but not nearly as much bravery as Rikki choosing to follow Nagaina into her den.
It was dark in the hole; and Rikki-tikki never knew when it might open out and give Nagaina room to turn and strike at him. He held on savagely, and stuck out his feet to act as brakes on the dark slope of the hot, moist earth.
Lastly, Rikki is loyal. The reader needs to remember that Rikki is not a human, despite how human he can act at times. He's a mongoose, and he doesn't owe the family anything. Rikki could leave the humans and the humans might be hurt, but they wouldn't question it. They know that Rikki is an animal. Rikki could leave, keep his own safety, and let the humans worry about the snakes. But Rikki doesn't do this. He stays fiercely loyal to his human family and is willing to lay his life down in order to save them. That's an incredibly human thing to do.