“Will they change the name to the Kansas Chiefs?”
Essentially Kansas City is a shared city, part in Kansas and part in Missouri. No name change required.
Surely they have different mayors?
Other examples: Bristol TN/VA
Bluefield WV/VA
There are plenty of cities in 2 or more counties.
Not really. I grew up in Johnson County, KS. Nobody went to Kansas City, Kansas (Wyandotte County).