This isn't true. At least with respect to "soundalike" see, e.g., Waits v. Frito-Lay 978 F.2d 1093 and Midler v. Ford Motor Co. 849 F.2d 460.
The famous person "owns" the sound of their voice, and the non-famous person does not.
Get wrecked peasants.
A lot of people look handsome, sound handsome and are even stylish. So some rich person can randomly claim some one looks closer or sounds closer to them and hence needs to forced to wear a mask to prevented from even talking?
So what happens next, when AI bots begin to sing, compose music, teach, paint or anything for that matter?
"Can Corporation hire Famous person, or Non-Famous person?"