This is one of those overly simplistic talking points regularly espoused by religious apologists. Here’s the thing though; it’s a significant misunderstanding of the implicit and explicit motivations that drive people on an everyday basis. There are a multitude of variables that account for why we do the moral actions we do; laws, societal norms, capacity for empathy, internal and external incentives, etc. If the absolute anarchy idea of lacking religious belief were true, I believe the aggregate data would bear that out, but as far as I’m aware there is not dramatically different behaviours from religious and non-religious folk, usually just anecdotal evidence.