Does religion have Monopoly on Morals?

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.

Religion does not have a monopoly on morals. There are plenty of moral atheists. The real question being asked is this: In the absence of any sort of higher power is there any real point to morality? Is the law meaningful if there is no lawgiver?

The sense in morality without a higher power is to be able to live in society with rules that allow the people to be happier on average.

I'd call myself more agnostic than atheist and I believe I have great morals. Everything is always a choice though, so I really think religion has nothing to do with it. There have been many criminals in history who claim to be very religious and still commit crime and sin. Then they just simply believe if they confess their sins they will be forgiven because that's what religion teaches them.

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".