It's not a higher level label, they simply don't consider themselves Inuit in the same way Canadians generally don't consider themselves "American".

That's not much a problem, usually a characterization is for use by others, you don't necessarily have to agree. Germans don't consider themselves living in "Germany" either. Or, names like "Balkans" are not necessarily preferable by those there, but that's life.

It's more as if you decided to call Austrians "Germans" too because you can't be bothered to make the distinction.

Well, we could have just as well. That's just like my second example of "the Balkan nations".

Lots of people living in Europe don't consider themselves europeans.

The problem there is that "American" has two meanings. Not that it's homogenizing.

Does "Inuit" have two meanings?

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.

Logically, being characterized at one level doesn't prevent categorization into finer or coarser levels. In reality, there is a finite amount of information that can be conveyed, and the level of characterization is often whatever you lead with. There are important trade offs to be made with fitting in with a larger group vs being characterized as the mixture of traits of said larger group.