Does religion have Monopoly on Morals?

But as far as the initial question: Does religion have a monopoly on morals? Which came first, a religion or a society? In terms of the roots of human civilization and the foundation of social norms it brought, we might not ever truly know. It could be a mixture of both for all we know.

Moral behavior predates humanity,
unless you're preparing to argue that wolves and elephants have religion.
Pack and herd animals all obey certain rules that enable the group to serve its purpose of mutual benefit.

No. If there is no person or entity of a higher authority to punish you, the law or moral code is meaningless. Because most humans see each other as equals, no human being can truly ever be placed in a higher authority position, even if that of a judge, governor, etc. because in a scientific sense... no human is greater than another.

The group is the "higher authority".
If you misbehave you will be cast out or lynched, by the people themselves.

Also, humans don't see each other as equals. Like any pack animal, we crave "leadership". Most people will blindly follow, obey, and worship another human being if they become convinced (for any reason) that he's a "great person".