It has twice been proposed in the Navajo Nation Council to officially change the name from Navajo to Dine and both times the proposal was voted down. This was largely because the proposal was unpopular among members of the tribe. There are multiple generations that have grown up being Navajo, they think of themselves as Navajo, and they want to continue to be referred to as Navajo.

Also, growing up (white) on the Navajo reservation, I was taught that Dine was the native language word for themselves, but no one ever expressed any desire for white folks to refer to them as Dine. They tended be private about their culture - there were aspects that were for them not for us, sometimes plainly and other times more subtly. Dine sometimes had that feel, like it was a private name amongst themselves, not a public name for outsiders to use. In other ways it was similar to German vs Deutsch, just a different word in a different language.

It is more recently that some Dine activists have cast off the name Navajo as colonial and wish to be referred to as Dine, but as far as I can tell it still isn't the majority opinion.

Bizarrely, Germans call themselves Deutsche and the Dutch call themselves all sorts of crazy things. Swiss, Spanish, oh the whole of Europe.

As for Japanese, Chinese, Russia ... errm most of the rest of the world 8)

In English we don't really get America(n) "right", and they are one of the larger populations that speak the bloody lingo! Oh and lingo: lingua - Latin with a twist, that's how we Anglo-Saxons keep it real!

"The Navajo are speakers of a Na-Dené Southern Athabaskan language which they call Diné bizaad (lit. 'People's language'). The term Navajo comes from Spanish missionaries and historians who referred to the Pueblo Indians through this term, although they referred to themselves as the Diné, meaning '(the) people'."

From: https://en.wikipedia.org/wiki/Navajo

English speakers don't call anyone by their own name. "Japan" is another good example here.

Interestingly, the Japanese term for other countries/languages are frequently much closer than the English versions. So while English speakers call the Deutsch "German" and Deutschland "Germany", Japanese speakers call them "doitsu", which is just as close as you can transliterate "Deutsch" to the syllables allowed in Japanese. It's not a rule though: the proper term for the USA is "beikoku", which translates to "country of rice". (The colloquial term for USA however is "amerika", which is exactly what Americans call themselves colloquially.)