I don't think it's that clear at all. You own your "likeness", but the limits of what that means is highly untested. Of the similar examples that have been tested in court thus far, the Ford v Midler case is the closest, but the court specifically called out the fact that as a singer her voice is a distinctive part of their identity, and so it is protected.

https://en.wikipedia.org/wiki/Midler_v._Ford_Motor_Co.