A significant number of what are treated as Christmas songs are merely winter songs. These would include "Winter Wonderland," "Let It Snow," "Sleigh Ride" (either composed by Leroy Anderson or Leopold Mozart -- yes I know the Mozart usually is spelled without the space) and probably "Jingle Bells." None of these songs have anything to do with either the secular Christmas Holiday or the sacred tradition behind it.
What is really odd about this is that for most of the Northern Hemisphere the real winter weather, either the type described in the songs, or the harsher or milder weather appropriate for the region, won't really arrive until after Christmas in January and February. so by the time that we could be out playing in a winter wonderland, or enjoying a sleigh ride, these songs won't be heard since they are Christmas songs, only appropriate for the month of December.
One other song that is often classified as a Christmas Song is "Good King Wenceslas," yet the lyrics clearly state that the events described take place on The Feast of Stephen, which according to the google search I just conducted is December 26.
Non-seasonal albums released by the Christian recording labels will contain very few songs without explicitly Christian lyrics. Often if an artist includes a few, or even one, song that is not recognizable to most Christians as having Christian content on an album, they will be accused of "selling out." On the other hand, non-seasonal albums released by the secular labels will rarely contain songs with explicitly Christian lyrics, although releasing such a song rarely causes any notice. However, Christmas records released by Christian recording tables, featuring their regular artists will often include many secular Christmas songs (and winter songs). Similarly, many secular recording artists -- including many who aren't Christian -- will record old Christmas Hymns and other songs with explicit Christian lyrics. I even know of at least one example of a song on a Christmas album that has explicitly Christian lyrics, but is not a Christmas song: "Little Alter Boy" released on the Carpenter's A Christmas Portrait. (The lyrics of this song reflects a theology of asking someone else to pray for your salvation, leading me to often want to tell the character singing the song that they don't need the Alter Boy, they could just pray themselves, but I digress)
A corollary observation is that secular radio stations will play Christmas songs by Christian artists who aren't normally on their playlist -- in 1991 I heard David Meece's "Almost Christmastime" on a secular station in Sacramento California. And Christian radio stations will play Christmas songs by secular artists -- this morning K-Love played Bing Crosby's recording of "White Christmas."
(On the other hand, I've heard of a case of "White Christmas" being played in June or July. Sometime in the 1980's, Albuquerque hit 100 degrees for several days -- which doesn't happen all that often. One day just after the temperature officially hit 100 degrees, the DJ on the radio station my brother was listening to celebrated this by playing "White Christmas." But that is a special case)