I have some opinions about the Christmas songs that never actually mention Christmas…Winter Wonderland, Jingle Bells (originally written as a Thanksgiving song), and Frosty the Snowman, to name a few. But I don’t think that’s what you’re asking.
Owing to its inclusion on the Charlie Brown Christmas special & album, I’d say ‘Für Elise’ probably fits the bill for a lot of people. Leonard Cohen’s ‘Hallelujah’ has also been (inexplicably!!!) co-opted to some extent in recent years, thanks largely to the Pentatonix. Lastly, I’d put ‘Ave Maria’ on this list. It has obvious connections to Christmas, but is used much more broadly throughout the church year, yet most people (including me) would never listen to it in casual contexts except at CHRISTMASTIME.
Haha, my phone autocorrects to CHRISTMASTIME and I’m not “fixing” that.