It's as if the cat used the power within them to save the human, but at the cost of their physical body and so with the soul/power within the human, that kept them alive after whatever near-death experience happened, slowly merges and takes over. Not out of the cat's will, I'd guess.
We saw the shrine maiden with cat characteristics but still humanoid-ish (as in human sized), but I do wonder what happens if someone doesn't stop the process at all... Do they turn into a hybrid (anthropomorphic cat, think furry OCs) or do they themselves become a cat? Because the power that saved the human's life is compatible with cats, not humans, so... Maybe the power demands a feline body to work.
Perhaps along with it, the human never dying as long as the cat's soul is inside could be a possibility. And if they fully turn into a cat, the human becomes a yokai or something...