How does this story pertain to religion?

While religion doesn't have an explicit presence in "Babylon Revisited," the theme of redemption is nonetheless prominent. Indeed, the whole story can be seen as a depiction of one man's redemption as he rejects the life of a hard-drinking party animal to embrace that of a sober, responsible businessman.
When Charlie used to visit Paris back in the day, it was like Babylon, the ancient city, which in the Bible is used as a metaphor for all that is sinful and unholy. Yet now, when Charlie returns to Paris after all these years, he finds the place in a completely different condition from the debauched fleshpot of old. Not only has Paris changed, but he's changed with it; he's finally seen the error of his ways and now understands the need to sober up if he's to be reunited with his daughter, Honoria.
In Christianity, redemption can be defined as the deliverance from evil for the payment of a price. The price that Charlie has to pay for his deliverance from a former life of dissipation is to give up alcohol. This might not sound especially difficult, but to a borderline alcoholic like Charlie it's no mean feat. To achieve deliverance and all that goes with it—taking custody of Honoria and doing right by her—he must give up the booze, no matter how hard it will be for him.