Do you think that Jekyll would have to resort to creating Hyde if he was in a modern setting? I can't see why as most circles in society, the highly religious not withstanding, are not as puritanical as back in Victorian England.
It’s true that the exact social environment of the Victorian period doesn’t exist in modern day, but the kind of social pressures are certainly still around. Even if you live in a society that is generally accepting and liberal, you may carry with you a set of repressive values from your childhood that control your perception of the world. Or you may find yourself in a small, suffocating environment of people who regularly scrutinize you and judge you based on superficial things (as can easily happen in a highschool, workplace, etc.)
If the story of Jekyll and Hyde still speaks to modern audiences, then it must hit some emotional chord with people in their own lives. If modern society were 100% different from the late Victorian era, Jekyll and Hyde would probably have faded entirely from popular memory. Instead it’s one of the most well-known and commonly referenced stories in existence. I think that says something by itself!
Nitpicky nerd thing, but I don’t think the actual text of the story makes much reference to religion. Jekyll seems much more concerned with social pressures than religious ones. He cares much more about the judgment of people around him, rather than judgement from God.