It's more like if you took American to just mean "people of the USA". So it's more like "Canadians don't consider themselves part of the people of the USA". Inuit refers to a specific subgroup of people which have it as a word in their language. Many of those not in it, if you apply the label to them, they view it as if you insisted Canadians were part of the US people because "they live in the roughly same area and live kind of similarly".

Part of it is the word Eskimo. Not popular and falling out of fashion, people look for a replacement, and grabbed what a large subgroup of the people called that calls itself. And thus mislabeling those that are not in that subgroup.

To stretch the analogy again, lets assume for some reason "North American" has become a word to avoid, so lets just call all North Americans what the people from the USA call themselves to be more considerate. And the Canadians aren't happy with that.