In fact, during the Era in which Discovery is set, starship life is pretty much exactly like serving on a submarine, albeit slightly more spacious, and with science labs.The one exception might be submariners as their life is pretty heavily regulated. But it's a safe bet that starship life would be nothing like that.
Also, since when were religion and culture mutually exclusive?
It's pretty clear that humans are not exempt from racism, and cultural differences, this runs through most Trek episodes, and minimal group paradigms are well known, well researched phenomena. Of course on a Federation starship, your shipmates are part of your group and so any racial tension is likely to be directed elsewhere.