YES!
One possible implementation of empathy in a game is cooperation ->
It can be deep, complex, simple, fun, easy to play hard to master, whichever kind of cooperation -> leading to more synergy, dare I say symbiosis between the players!
Ok, random idea that just popped in my brain:
You could have a cooperative game where the goal is to handle nutrients, etc. in order to cooperatively build a baby inside's a female womb.
Basically, it would be about achieving "life" by cooperating: repelling microbes, driving whatever fluids/vitamins/hormones are needed to the right places, etc. etc.
Human embryos are actually very aggressive, they breach the mother's body and steal everything. Before implanting the embryo has to convince the mother that it is viable by a trial of strength in a hostile environment, the endometrium, which has an auto-destruct button the mother's body can push to reject the embryo. If accepted the embryo and the mother starts a nine month tug-of-war. They bombard each other with hormones, they try to suppress each other. Behind all of it the conflict is between the paternal and maternal genome. If they both prove to be capable fighters they both get what they want, a healthy baby and mother. If one side gets the upper hand things go wrong.
This is why pregnancy is so perilous. It's a war, presumably because human development requires a lot of resources, and the father can always find a new woman to impregnate, so it's genes best served by trying to steal said resources, as long as most women most of the time are capable defending themselves until the baby is born, preferably longer things chug along.
Not as complex as you are suggesting but in an older FPS they simply added medic ability to all players, so that any other player can heal another if they switch from shooting to that mode by simply pressing a button (delaying ability to switch back to a weapon)... Even though it's incredibly basic, it created a different sense of value among team mates that merely another shooter does not.
Real empathy is hard and the real world could stand to see more of it.