On the other hand, there was no keen intimacy between the dog and the man. The one was the toil slave of the other, and the only caresses it had ever received were the caresses of the whiplash and of harsh and menacing throat sounds that threatened the whiplash. (16)
The dog doesn't care about the man's well-being. It's completely self-interested and it wants to be near the man for his food and fire. Also, it has been taught to fear the man's whiplashes. Sounds more like a slave than a companion.
Now the tree under which he had [built his fire] carried a weight of snow on its boughs […] High up in the tree one bough capsized its load of snow […] It grew like an avalanche, and it descended without warning upon the man and the fire, and the fire was blotted out! Where it had burned was a mantle of fresh and disordered snow. (23)
Nature seems to always be working against the man's best efforts to save himself. At first, it might seem as though nature is just plain out to get the poor guy. But the use of the word "disordered" to describe the fallen snow suggests that there is no rhyme or reason to what's happening. It's just randomness mixed with really harsh surroundings.
And all the while the dog sat and watched him [build the fire], a certain yearning wistfulness in its eyes, for it looked upon him as the fire provider, and the fire was slow in coming. (25)
This quote foreshadows the story's ending, suggesting in its first half that the wolf dog feels bad for the man, then undercutting this point by saying that the dog just wants fire to warm itself.
Later the dog whined loudly. And still later it crept close to the man and caught the scent of death. This made the animal bristle and back away. A little longer it delayed, howling under the stars that leaped and danced and shone brightly in the cold sky. Then it turned and trotted up the trail in the direction of the camp it knew, where were the other food providers and fire providers. (42)
Here, we seem to have a moment of tenderness between the dog and man. Confronted with the man's death, the dog howls out of sadness into the deep cold space of the night sky. The poetic images of leaping and dancing stars suggest a sort of cosmic beauty in the northern sky. But then the dog turns and casually "trots" away in search of other men who can provide food and fire. It has no special bond with the man, but just sees him as a failed provider of fire and food.