Of course the "belief" of right and wrong goes back.
But for simplicity, let us suppose someone does not believe in God, etc.. And they say "Now that's wrong".
Okay ... explain why? What makes it wrong. Based on what? Thin air? Prove its wrong? Explain why something is right? What are the mechanics of "right"?
Never seen anybody prove it. It's a belief system. Like you wrote to basically.
And a belief system is pretty much a religion. It does not take a G,O,D to make a religion.
So if one person wants to remove God from schools, or Buddha from schools, another person can honestly request that "right and wrong" also be removed from school. Many folks wouldn't like such a request, but it would be a fair request in a culture that claims religion and government should be separate.
What you wrote about "right" going back a long time for beliefs, fits what I believe about God and mankind.
From a believer's point of view, I believe that in the origin of man (created by God), that all men had the purity of right and wrong. As all went their separate ways and formed nations or clans, each group retained portions of the teaching.
To your other point, assuming we were created, God did not use Homo sapien. No need to classify. We know who we are. The classification is a need for people feeling we need to be pigion-holed into the animal kingdom.
Scripturally, man has mainly been separated into classifications of nationality, religion, etc.: Saducees, Baal Worshipers, Hitites, Egyptian, Caananites, Hebrews ...
And believer or unbeliever too. But I'm not familiar with any counterpart for Homo sapiens, though I may have missed it.