>There are thousands of indigenous groups in the Americas and every person in them can have their own preferences and opinions.

Well, the same is the case with Europeans, Asians, and so on, so?

We have smaller identities, and larger group identities, it's not belonging or being categorized in one precludes the smaller more unique identity...

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.

Eurasians sounds like a neat way to group people who are given a handicap when they apply to Harvard.

I have never heard anyone taking offense to the word Inuit. It simply means "people" in Kalaallisut.

Greenland is quite far from places like Utqiagvik and Chukotka, where indigenous locals speak entirely different languages. Is it any surprise some people wouldn't want to be labeled that way?

I'll admit "inuit" is pretty widely accepted nowadays, but it's not universally so. The relevant ANLC page gives a good summary of the situation: https://www.uaf.edu/anlc/resources/inuit_or_eskimo.php