Perhaps they only have proper carols in regions with more Legendaries to sing about? Granted, it does seem like it would be more likely in a region with a more prominent Ice Legendary or a Legendary more explicitly linked to the sun. Christmas is likely some sort of solstice festival here, and I'm sure the name must be some sort of strange coincidence.

[can you even rhyme articuno, regice, or kyurem in a way that isn't stupid

....and... yeah...]

It really does, doesn't it? I don't know what to make of it. I doubt it's a Legendary of any kind, but it's hard to say what it actually is.