James Pond II: Codename Robocod (MD/Genesis) sees the main villain take over Santa's Workshop and is super Christmassy themed right from the start. I was never a huge fan of the James Pond games so I don't know how deep into the game the theme runs, but my understanding is that it does. I think the game itself has a good reputation too from what I've heard.
There's a few games based on movies that, whilst not strictly
Christmas films, are still Christmassy. Like
Batman Returns (MD/Genesis, SNES),
Home Alone 1 & 2 (MD/Genesis, NES, SNES). I can't speak for the quality of any of them (although I do remember Home Alone 2 getting ripped into by AVGN), but the first Home Alone game on MD/Genesis has you placing traps in houses around Kevin's neighbourhood to foil the Wet Bandits. I've only played it super briefly, so I'm not sure if the homes are decorated for Christmas, but when traveling between them Kevin rides on a sled through snow covered streets; I've heard some good things about it. I guess there's also
Die Hard Trilogy on PS1 but I don't remember any of the Christmas stuff carrying over.
Following the obvious Winter Sports theme, two games from my childhood on the Mega Drive are
Lillehammer '94 and
Winter Challenge, and were both kinda frustrating in their own unique ways, but Winter Olympic games were sparse back then! Another obvious choice are Snowboarding games –
Cool Boarders series on PS1 (I had the fourth one and was never a big fan, but it was ok back then) and the
SSX series (SSX Tricky on PS2 remains one of my favourite games ever). One of my other favourite winter sport games that I rarely hear get talked about is
Sled Storm on PS1; maybe it was the uniqueness of a snowmobile racing game back then, maybe it was the remixed Rob Zombie soundtrack, or maybe just cos I won the game in a Kellogg's Corn Flakes random prize draw and didn't expect it to be good, but I really enjoyed it
Going with just the winter theme in other games, there's the Bomberman-like
Pepenga Pengo (or just Pengo) on MD/Genesis where a penguin replaces Bomberman and ice blocks replace bombs. Its on my list to play someday and I've heard decent things about it. Also on Mega Drive is
Scooby-Doo Mystery, it's a point and click adventure/mystery game with two scenarios, one of which takes place in a snowy hotel/cabin/lodge; it's another I plan to check out simply because the art style is exactly like the cartoons but I've heard mixed things about it.
I've just realised the next two games I got for Christmas on PS1 in '99, which may be swaying me a bit, but dammit I personally still count them, regardless of that bias!
First is
South Park, which, yeah isn't great and is especially not great if you're not a fan of the show, but like the show everywhere is covered in snow. Second is
Mission: Impossible which is bookended by missions in Siberia (I think) but, to be fair, I think the entire middle of the game has nothing remotely wintery about it. Its either my own seasonal nostalgia, or the first and last missions being snowy that convince me the rest of the game is taking place during winter. It also doesn't have a great reputation, but I enjoyed it at the time. Arguably the N64 versions of both are slightly... less-bad.
An obvious PS1 choice is
MGS, even if the moments in the snow are limited.
I can think of a few other games that had the occasional snowy or Christmas level, but I'm not sure if you're after games with just an occasional winter level?
Totally cool idea for gaming over the holidays, though
I usually get my Christmas gaming feels from games/demos I remember playing around winter/Christmas. Or movie-license games from films that I either saw for the first time at Christmas or that were commonly on TV at Christmas. So like another thing about the Mission: Impossible game was, coincidentally, that very same Christmas I got the game, the movie was on TV on Boxing Day (and being the BBC, they promoted all their upcoming movie over Christmas constantly until the New Year, so the promo for it was driven into my brain every day for a month) So a similar feeling happens to me with the Alien movies/games, and the original Italian Job always seemed to be on TV around Christmas in the UK so naturally the PS1 game used to feel like one to play around then.
Are there any other criteria along those lines that might broaden the scope a bit? I never realised how few options there were until I properly thought about it