That's not weird at all! I'm honestly quite pleased with that aspect, because during early development snow wasn't a part of it. While I was looking through my huge collection of photos, I kept finding some snowy ones and Christmas ones, and my teammate on the game said "We should use some of these snow photos!"
So I dug through and found a bunch more, then we added the snow effect, and I realized I had to set the game around Christmas. The game's title didn't come until about two weeks before the end of the jam!
At the time I was writing, I was listening to a lot of Dir en grey, and it hit me to use the name "Kasumi" or "haze/mist" from one of their songs. And I kept listening to "Ain't Afraid to Die" which is a song about missing a lost lover on the last snow day of the season... and yeah, it slotted right into the story so perfectly! That's when I decided to put lyrics into the story. And then a couple days before the end of the jam, I struck on the idea of doing an SNES audio cover of the song, so that's what plays in the big reveal scene and the final scene (Christmas evening).
So yeah, (surprisingly!) completely unrelated to Banshee's Last Cry, snow became hugely important to the story. I'm happy it resonated with you!