I believe it is possible societies may have morality without religion, there is most likely no monopoly by religion on morality. However it is hard to see what that morality would consist of. Buddhism gives us conpassion to all beings, Christianity gives us the golden rule--what kind of morality wouldn't consist of those. Not that all religion is good, Islam teaches death to Jihadists. Now you may ask those first two are so simple anyone could have thought of those. I say it took 1 billion years of evolution to understand them so they must be pretty important. Its easy to stand back from religion and poke holes in it, but if you immerse yourself in it properly you may see Isaac's son was plotting a murder and that therapist is a quack. Oh, and that the golden rule was discovered.
That means that we came up with morals without religion and even came up with morals despite religion. Infact i would argue that not only could we practice a sufficient modicum of morality without religion but we could progress morally much faster. Every moral progress in the world has been opposed by the organized churches of the world. The diminution of war, the mitigation of slavery or every step towards the better treatment of colored races, every improvement in humane feeling, every improvement in criminal law, and much more progress has been consistently opposed by the organized churches of the world. So it would seem that we couldnt practice a good rational modicum of morality without ignoring all religions we have today. Its far outdated and its irrational to draw your vision of how to live in this world from ancient literature when todays therapists would diagnose the people who wrote this crap with mental illnesses and when our judicial system which we like would lock all of them up . If you go to your therapist and say god wants me to sacrifice my child to him they are going to call the cops and tell them some psychotic person is planning on killing his or her child. Also if god tells you to rape and abuse women and that women should have no rights like they used to so i can do whatever i want to women according to god they will consider you psychotic.