Morality by another name is ethics. By definition, it is knowing what is right or wrong, good or bad.
Mammals, whether human or not, seem to instinctively know what is good or bad. Good helps them survive and procreate. Bad causes pain, death, or other discomfort.
The psychological aspect of what is good or bad is tied to empathy. You know that biting hurts, so you refrain from biting when you don't wish to inflict pain. It might take socializing with others before you getting bitten teaches you that lesson, but you eventually do learn it.
One way to describe empathy is in the Bible. When asked which of the 10 Commandments is the greatest, Jesus replies, "Here is all you need to know. Do unto others as you would have them do unto you."
That pretty much spells out what we as living beings know WITHOUT THE NEED for a "construct." We innately and instinctively know good and bad because we know what we would like to have done to us, as well as what we don't want done.
Therefore, in my opinion, good and bad, right and wrong, ethics and morality all exist in us without our need to have a standard defined for us. For instance, if stealing is wrong, what about stealing to eat and survive? These are ethical dilemmas the law has struggled with for ages, which is why we have judges and juries. The assignment of GOOD and BAD to individual behavior is not so simple to do on paper.
Those who lack a basic ability to empathize are what we often refer to as "psychopaths", and those who know better but choose to do the opposite are "sociopaths". As a whole, society accepts and conforms to the behavioral norms.they understand as acceptable.
Also in my opinion, I believe it is a relatively recent development that people have come to trust that anything that is wrong is already illegal. And anything that is not specifically forbidden by some law or guidance is permissible. That is a cop-out. Just because something you do is legal doesn't mean it is the right thing to do.
I think long before there was organized religion or criminal courts, there were standards of behavior expected of everyone in a given societal setting. Those standards have changed over time, and they have been codified as "Commandments" and "laws", but 98% of us know when we are doing something wrong even when there is no formal prohibition against it.