The areas that form the Assyrian homeland are parts of present-day northern Iraq, southeastern Turkey, northwestern Iran and, more recently, northeastern Syria.
I think it's like Iranians that call themselves Persian because they don't want to be associated with the current regime/country.
The group has 7,000 years of history, and like Zoroastrians, predate Islam.
Yhey too suffered a Holocaust, like the Armenians.
I recently me a woman, from this group. Some of them speak Armaic, the language of Jesus and of the Bible.
So yes, it is a "thing"