Yes, it seems like there was or is a region considered the "Assyrian homeland" [0] of the people for whom the empire was named (Assyria being named for the home city of Assur). Wikipedia's map makes it look the same as the Kurdish territory and when I look up differences between them, Reddit threads describing contemporary accounts are front and center. [1]
0. https://en.wikipedia.org/wiki/Assyrian_homeland
1. https://www.reddit.com/r/Assyria/comments/u8c324/relationshi...
> Assyria being named for the home city of Assur
Well, sort of. "Assyria" would be a rendering of the Greek idea of the name. The Greeks couldn't pronounce it.
In English the city (and god) is usually called "Ashur"; in Akkadian it's Ashshur. It's never called "Assur".