It's not unique to mammals, BTW. Some cuckoos and cowbirds are "brood parasitic" and make use of the maternal instinct of other birds. But "maternal" is not strictly correct, as some of the care-givers are male. Among the recent videos I saw was of (IIRC) a MALE pit bull that had a baby kitten - carried it around in its mouth all over the place.
In a sense, humans make use of this too - with the animals we take in as pets / partners. When our dog dies, I'm going to have to put my wife on a suicide watch. That's not a joke. I'm serious. I've already broached the subject with her several times so she knows this is an expected outcome and something we can deal with.
But humans also benefit from this - there was a case a few years back when a kid in Germany (I think) who fell into a gorilla cage and was protected by a female who dragged the boy over to the door and kept the male gorilla away. There are several cases in recent years of dogs that died to protect their owners or displayed loyalty in some other way. (If you're interested, google "Hachiko", not the movie the real story of a Japanese dog.)
Those of us who were adopted, in a sense, make use of the same principle. (I was legally adopted by my step father, but it seems very odd to me and I usually don't even think about it until someone mentions my "real dad.") In any case, my brothers and I have benefited from our step father's need to nurture.
Good luck on finding the term. My only contribution is:
1) Humans display this character and also benefit from it sometimes, both with respect to other animals and with respect to ourselves.
2) The character is displayed somewhat in birds. I don't about other non-mammals. Monotremes? Reptiles? Fish? Are there limits even within mammalia?
I get slightly irritated when people say, "Oh, your dog is not experiencing and conveying love, but just programmed behavior."
What? It's not "real" unless it's also magical?
It reminds me vaguely of another programmed behavior - that of the Sphex wasp
http://en.wikipedia.org/wiki/Digger_wasp