Caring for animals has shaped our evolution as humans. No other species has domesticated a large variety of animals, fish, birds reptiles or insects. Most of us now need that animal connection.
And I know what you are saying FF, with regards to your wife's dog. I think we get closer to our animals than most other humans.

I think 'nature' made babies and toddlers of all animal groups, cute, so we would care for them. I should tell you that once I found a nest of rats in my bathroom and started leaving food for the babies, only to have someone comment that in a few months time I might be wanting to get in the exterminator........

As to your question H L ...I don't know of the word. There should be one. Good question. (and yes welcome, as Luke says).

alloparent or allomother....... using the Greek prefix, allo 'other than'.... describes an individual other than parents that help raise offspring,
as in some societies, both animal and human, where 'aunts' traditionally take on that roll.

I agree that the word shouldn't be 'maternal' as in female. Many males species are involved in raising young, some exclusive to the mother who delivered them. I'm thinking about seahorses now, but thats more role reversal, because the male gets 'pregnant and delivers the young fry' and neither parent rears the young....not a good example.