If you'd like this breadcrumb as something that could help make the world of the story feel like it could be our world, here's an interesting thing which is that Alan Turing thought that telepathy could be possible, and tried to account for this when designing the Turing Test.
If telepathy were to be possible as described in your story, you could explain it as mind-body dualism being real https://en.wikipedia.org/wiki/Mind%E2%80%93body_dualism and that the new animal born from the person temporarily shares a neural link with the person's brain, causing the sharing of one spirit, and that when the neural link is broken as the animal separates, the spirit is still anchored to both parts of the formerly-unified nervous system. The explanation for why this wouldn't happen during standard childbirth is that mother and child don't share a neural link, only a blood link.
Your telepathy explanation is exactly where I was coming from. This is why I believe we'll be able to live forever without uploading our minds into computers.
