The reason beasts give among themselves [for not hunting man] is that Man is the weakest and most defenceless [sic] of all living things, and it is unsportsmanlike to touch him. (1.19)
This is part of an unspoken moral code among the animals, and it stands in contrast to Shere Khan, who has no honor, and has no problem snacking on any kids who cross his path.
"By the Broken Lock that freed me, I am sure, Little Brother." (1.94)
Bagheera invokes this image of his imprisonment and subsequent freedom as a symbol of his honor, kind of in the same way Liz Lemon says, "By the Hammer of Thor!"
Akela lifted his head again, and said: "He has eaten our food. He has slept with us. He has driven game for us. He has broken no word of the Law of the Jungle." (1.122)
Akela tries to remind the Wolf Pack of their principles, but they still betray Mowgli by wanting him gone, simply because he is different.
"I can save ye the same that comes of killing a brother against whom there is no fault—a brother spoken for and bought into the Pack according to the Law of the Jungle." (1.127)
Aside from Akela, Mother Wolf, and Father Wolf, the rest of the wolves don't really have any honor. They almost all secretly want to be a lone wolf on the inside, so they sabotage their own pack's leadership.
"Run back, Messua. This is one of the foolish tales they tell under the big tree at dusk. I have at least paid for thy son's life. Farewell; and run quickly, for I shall send the herd in more swiftly than their brickbats. I am no wizard, Messua. Farewell!" (5.98)
One of the reasons Mowgli kills Shere Khan is to avenge the death of Messua's son—Messua being the woman who took him in.
"No; we will not hurt the village, for Messua was kind to me." (5.102)
As much as Mowgli might want to destroy the village, he doesn't, instead returning the kindness Messua has shown him.
But none of the other seals had seen the killing, and that made the difference between [Kotick] and his friends. (7.61)
Kotick sticks with his desire to save the other seals, merely because it's a nice thing to do. None of the other seals seem to care that a few dozen of them get killed every now and then, but Kotick is still determined to stop it.
Kala Nag would no more have dreamed of disobeying his shrill little orders than he would have dreamed of killing him on that day when Big Toomai carried the little brown baby under Kala Nag's tusks (11.7)
Kala Nag is basically enslaved, and he knows that he can't hurt Little Toomai because the consequences would be severe. Do you think Kala Nag would act differently if he were wild?
Toomai knew that, so long as he lay still on Kala Nag's neck, nothing would happen to him. (11.62)
Toomai knows he can trust Kala Nag, and we have a feeling that Little Toomai will treat the elephant well (by which we mean as well as a slave animal can be treated) when he becomes the mahout in charge.
"They obey, as the men do. […] Thus it is done." (13.145)
It seems that everyone, whether human or animal, obeys the orders from their superior because it is honorable to do so. To disobey and run would be disrespectful.