My second oldest has said there are five basic categories of Christmas Song.
There is the historical secular, the old traditionals. You know, Boar's Head Carol, Here We Come A-Wassailing, We Wish You a Merry Christmas, Deck the Halls, and so on. Some of these date back centuries and remind us that Christmas has always vied with the secular celebrations of Winter. It's not a new conflict.
Then there is the religious songs of all ages. Silent Night, O Come All Ye Faithful, Angels We Have Heard on High, O Holy Night. No matter what the age or century, the general subject matter of the songs are the same: Christian celebration and reflection plain and simple. The list of these songs is long but, significantly, confined largely to a pre-WW2 time period.
Then comes the 'Classical' Secular. Jingle Bells, Sleigh Ride, White Christmas, The Christmas Song, Here Comes Santa Claus, Winter Wonderland. Some of these date as late as the 1960s, and stretch back to the 19th Century, but generally don't include the ones earlier (see Historical Secular above). They are also mostly confined to the 1940s or 1950s or before. Many were written in the context of the Americanization and commercialization of Christmas. This is likely the fuzziest category. After all, is Rockin' Around the Christmas Tree in this or the next category? How about Jingle Bell Rock?
Next is the Modern Secular. I subdivided this into two groups. The MS Frivolous, and the MS meaningful - but not Christian meaningful. In the first group is Feliz Navidad, All I Want for Christmas is You, Wonderful Christmastime, Merry Christmas Darling. Clearly Christmas songs but nothing more about the manger than Silver Bells. In the second is such fare as My Grown Up Christmas List, Do They Know it's Christmas and, most famously, Happy Xmas (War is Over). Songs that use Christmas to promote some meaningful message - but one shorn of Christian dressing.
Finally comes the secular non-Christmas. That is modern - meaning largely post-war - songs that have nothing to do with Christmas at all. They may not even mention winter. They could take place in August or March for all it matters. Sometimes they had nothing to do with Christmas but became associated with the holidays for this or that reason. Think Baby It's Cold Outside or Last Christmas.
The last one brought us to the video above. My son pointed out that the song could change its name to Last Arbor Day and not a speck of meaning would be changed. At least Grandma Got Run Over by a Reindeer mentions Santa. On the other hand, some of the classics, like Jingle Bells, are no more beholden to Christmas than Wham's little ditty. And perhaps in a hundred years it will be as engrained in the cultural mindset as Jingle Bells or Winter Wonderland.
But for now, that's a breakdown I can accept. And sometimes it might be a matter of how the song is presented as the song itself. After all, hearing it in the medieval style makes it seem almost plausible as a Christmas tune. Almost.