Organized religions certainly have their own standards for morality, yet morality is not something that is confined to religious individuals. In reality, however, it would seem to be somewhat more complicated than that.
Morality seems to exist on several different levels.
Individuals have their own personal morality. Small groups have their morality - which may well differ from the views of the individuals within the group. As you combine people into larger and larger groups or change the yardsticks by which you group them, you find a different set of moralities for each different group. I think we can see the answer in the various levels of our judicial system - both formal and informal. Disputes are normally resolved at the lowest group level according to that group's moralities.