That is what matters. OWNERSHIP over her contributions to the world.
If someone licenses an impersonator's voice and it gets very close to the real thing, that feels like an impossible situation for a court to settle and it should probably just be legal (if repugnant).
This is a civil issue, and actors get broad rights to their likeliness. Kim Kardashian sued Old Navy for using a look-alike actress in an ad; old Navy chose to settle, which makes it appear like "the real actress wasn't involved in any way" may not be a perfect defense. The timeline makes it clear they wanted it to sound like Scarlett's voice, the actual mechanics on how they got the AI to sound like that is only part of the story.
They literally hired an impersonator, and it cost them 2.5 million (~6 million today).
https://www.latimes.com/archives/la-xpm-1990-05-09-me-238-st...
Because then the actual case would be fairly bizarre: an entirely separate person, selling the rights to their own likeness as they are entitled to do, is being prohibited from doing that by the courts because they sound too much like an already famous person.
EDIT: Also up front I'm not sure you can entirely discuss timelines for changing out technology here. We have voice cloning systems that can do it with as little as 15 seconds of audio. So having a demo reel of what they wanted to do that they could've used on a few days notice isn't unrealistic - and training a model and not using it or releasing it also isn't illegal.
Does that mean if cosplayers dress up like some other character, they can use that version of the character in their games/media? I think it should be equally simple to settle. It's different if it's their natural voice. Even then, it brings into question whether they can use "doppelgangers" legally.
If the voice was only trained on the voice of the character she played in Her, would she have any standing in claiming some kind of infringement?
She doesn't have to own anything to claim this right, if the value of her voice is recognizable.