A person's voice is effectively owned by the corresponding person through right of publicity, which includes voice depending on jurisdiction.
California, for example:
"Any person who knowingly uses another’s name, voice, signature, photograph, or likeness, in any manner, on or in products, merchandise, or goods, or for purposes of advertising or selling, or soliciting purchases of, products, merchandise, goods or services, without such person’s prior consent, or, in the case of a minor, the prior consent of his parent or legal guardian, shall be liable for any damages sustained by the person or persons injured as a result thereof."
Voices can sound very similar, they're far from unique. Clearly if you say or somehow strongly imply that a voice belongs to a specific person then that is protected. But what if you use someone's voice, someone not especially well known, and don't make any claims about where it comes from?
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.
California, for example:
"Any person who knowingly uses another’s name, voice, signature, photograph, or likeness, in any manner, on or in products, merchandise, or goods, or for purposes of advertising or selling, or soliciting purchases of, products, merchandise, goods or services, without such person’s prior consent, or, in the case of a minor, the prior consent of his parent or legal guardian, shall be liable for any damages sustained by the person or persons injured as a result thereof."
https://leginfo.legislature.ca.gov/faces/codes_displaySectio....