There are two main reasons the narrator of this story is omniscient instead of limited: first, the narrator not only tells us what the man is thinking, but contrasts it with what the dog is thinking, like in the following quote: "[The dog] knew that it was no time for traveling. Its instinct told it a truer tale than was told by the man's judgment" (6). Burn. Take that, unnamed man.
Second, the narrator shows its omniscience by actually stepping in and judging the man pretty harshly. It's not as bad as what you sometimes find in high school hallways, but still pretty judgmental. We get a clear idea of this in paragraph three, where the narrator tells us that "The trouble with [the man] was that he was without imagination. He was quick and alert in the things of life, but only in the things, and not in the significance" (3). The story would be quite different if the narrator didn't butt in on our reading like this. We might actually sympathize with the man a lot more. Instead, we get this pushy narrator telling us not to get too attached to this guy because he just might be a bit of an idiot.
The narrator is also the only thing that gives us an insight into the difference between the man and the dog's responses to the cold weather and nature in general: "To permit the ice to remain would mean sore feet. [The dog] did not know this. It merely obeyed the mysterious prompting that arose from the deep crypts of its being. But the man knew, having reached a judgment on the subject" (13). Passages like these show us why humans can be pretty pathetic compared to dogs, since while the man can "know" things, there's only so much he can be aware of through experience. The reason he winds up perishing in the arctic tundra is because when it comes to traveling in seventy-five below zero, you might only get one shot at survival, with no experience to rely on. Only instinct can save you.