There is a moment in The Jungle Book in which Mowgli reaches up and under Bagheera's chin. There Mowgli discovers that Bagheera has a bald spot. The panther explains that nobody in the jungle knows that Bagheera carries that mark. It is the mark of a collar, showing Mowgli and readers that Bagheera had been raised in captivity by man. Bagheera freed himself, and he states that he is far more dangerous than Shere Khan because Bagheera knows the ways of men as well as the ways of the jungle. Mowgli then states that everything in the jungle—except for Mowgli himself—fears Bagheera.
“Yes,” said Mowgli, “all the jungle fear Bagheera—all except Mowgli.”
One reason that Mowgli doesn't fear Bagheera is because Bagheera has never given Mowgli a reason to. Bagheera has been one of Mowgli's primary caregivers and teachers since Mowgli first arrived in the jungle and was taken in by the Wolf Pack. Bagheera, the wolves, and Baloo are all fierce predators in their own right, but none of them have ever done anything to Mowgli that would cause him to begin fearing them. Additionally, readers repeatedly see that Mowgli is a boy without fear. It is one of the first things that Mother Wolf notices about him.
“He came naked, by night, alone and very hungry; yet he was not afraid!”
Mowgli will be asked time and again if he is afraid of something or someone, and Mowgli repeatedly responds by asking why he should be afraid. On the one hand, Mowgli could genuinely not be afraid because he is that brave. On the other hand, it could mean that Mowgli is simply too naive and doesn't grasp that he should be afraid. Overall, Bagheera has never given Mowgli a reason to fear him, and Mowgli is not characterized or controlled by fear.