How do we know this? I suspect that many of the higher creatures - whales, apes, some dogs, cuttlefish even ... have a combination of problem-solving skills, abstract capacity, memory and opinion to qualify for understanding, and acting on, right and wrong. I greatly doubt it's a human monopoly.
Regardless, "a sense of morality" is a philosophical concept, but "morals" is not. It's a codeword used by dominionists and related psychoterrorists for justifying an awful ideology. You preach, and yet you squirm to deny it. Interestingly, the NT warns that just those are the lukewarm sort who'll be spat out. cn