I don’t think you can enforce copyrights on someone sounding like someone else. You obviously can’t claim that it’s Drake, but I’m sure I can release a rap song where I rap like Drake (as many artists already do).
This is a PR stunt, it will never go to court anyway. But if it did, the way to have done this legally would have been the Death Row Records approach when Snoop Dogg left them in 1996. They simply signed a rapper who sounded like Snoop, gave called him Top Dogg, and had him perform guest verses for other label acts. He appears on 2pac's "All About U", recorded at a time when both Dr. Dre and Snoop had left the label.
What they *might* be is protected by other laws. The closest case the US has seen is this Bette Midler one: https://en.m.wikipedia.org/wiki/Midler_v._Ford_Motor_Co.
But the circumstances there make it different from a lot of AI voice work since in her case there could be no claim that sounding like her was a coincidence since Ford had an impersonator sing one of her songs (which they did properly secure copyright to use).