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.
Very nice, sort of reminds me of the Azabo from the Book of the New Sun. There is also a detach-3, but couldn't find any more story, nor an index to these stories in the website.
Very good investigation. Detach 4 wasn't good enough to upload but 3 did kind of expand into 5, 5.2, and 5.3. As much as I want to turn Detach 5 into a nine chapter book, it's not easy.
The original notes that inspired all this are at detach.html without suffix. None of these are linked on the website as they are a work in progress. (Why have a personal website without some hidden gems?) The dream I think would be an anthology of short stories related to different paths from this kind of multiplicative ability.
http://latenightsketches.com/detach-2.html