Upvote:3
In western culture, Red is the color of passion and it's disordered vice of lust is cooped up in the color. (see also, the Scarlet letter). I see the esteemed Revered V says the sin is pride. I don't disagree, I just think the real sin that is being covered here is lust and not pride. Karen has a disordered desire to dance and be seen when she should stand still and keep quiet.
As far as the particular religion is concerned. I don't see it being Catholicism:
She went to the parsonage, and begged that she might be taken into service there.
Parsonage is a is a Protestant term, now, either he's just using the wrong word there, which is understandable or he's referring to someone else, if it were Catholic, we'd call it a Rectory, not a parsonage. I don't think he didn't wrote deliberately Catholic stories like the Grimm brothers did, but I'm not an expert and have read more Grimm's Fairy Tales than his.
So, my answer would be, the same type of Christianity that Nathanael Hawthorne portrays in The Scarlet Letter, good old fashioned puritanism.