To even things up a little, some of these Hannukah songs seem to be by people who aren’t Jewish
Reading the lyrics, I don’t see how it could be about anything other than Christmas. There’s a star, a lamb, a shepherd boy and so forth. All very traditionally associated with the event.
Is there anything actually about Christmas in that song? I mean, in a religious or historical sense?
The lyrics are allegorical.
Even with the explosive popularity of “Do You Hear,” which especially took off after Bing Crosby recorded it in 1963, few people thought of it as anything other than a standard Christmas song. And the mention of “the Child” certainly places it within a Christmas context. There are clues in it, though, if you listen for them, that it has a contemporary framework, in particular the plea for people to “pray for peace . . . everywhere.” I had wondered vaguely if that line was a later addition to a standard text, when it is of course the whole point of the lyrics. And the line about the star with “a tail as big as a kite” has also puzzled me, but I figured it was simply a reference to the Christmas illustrations we see in which the lower ray of the star of Bethlehem is lengthened until it actually touches the roof of the stable. When you know the context of the words suddenly that “tail” becomes more ominous: it indeed evokes the biblical star that led the Magi to the Christ child. It also evokes a nuclear missile. It’s almost as if the song is asking, “Which one will it be?”
Don’t move the goalposts. The OP was talking about songs specifically about the season or mentioning it, which I[ve restricted myself to. “White Christmas” definitely falls into that category. That it doesn’t go into the religious significance is irrelevant.
Fair enough.
The answer for most of these songs would be that, even if they are not specifically tied to the holiday itself, they are evocative of the season and can thus be connected with it. “Winter Wonderland” conjures up lovely warm images of enjoying winter, and thus contributes to the atmosphere leading up to Christmas. Creating this atmosphere is probably part of the commercial exploitation of the season by merchants. The March of the Toy Soldiers may come from “Babes in Toyland”, but it could evoke the toy soldiers in “The Nutcracker”, which is Christmas-themed, or of toy soldiers guarding Santa’s workshop in a mall Santa display, etc.
On top of this, Christmas is not just the one day commemorating Jesus’ birth. It is a period of celebration (where do you think the title of the song “The Twelve Days of Christmas” comes from? Today, when we don’t live according to the rhythm of church holidays anymore, such considerations may be lost upon us). So the holiday of St. Stephen, which falls on what we term today Boxing Day, is relevant. It’s the second day of Christmas and part of the season of celebration (incidentally, in the Czech Republic, where St. Wenceslas was a duke back in the days when the Bohemian or Czech state was still forming, the 25th is considered the “first Christmas holiday” and the 26th is the “second Christmas holiday”, although people today have the main Christmas celebration on Christmas Eve).
So in conclusion, a lot of these songs are simply selected to contribute to the general winter atmosphere that supposedly makes Christmas (in its more secular and commercial modern incarnation) magical. Some also may allude to toys and other things associated with the season, even if the song itself is not directly Christmas-related. Given that the holiday has lost its religious significance for many people, those who select this music don’t feel the need to be boxed in and only play “Come All Ye Faithful”, “The Coventry Carol”, or other overtly religious Christmas songs.
But, as I point out, both the operetta and the first movie version are themselves Christmas-themed, as well. March of the Toys doesn’t need a secondary association with The Nutcracker in order to be associated with Christmas*
*As I Say, March of the Wooden Soldiers appears to be a different and earlier work, but people invariably call the one from Babes in Toyland “March of the Wooden Soldiers”. Probably because that’s what they titled the 1934 movie. .
“It’s A Marshmallow World” is another generic winter song that is almost never heard after Jan 1.
Huh. Is “Milord” also a Christmas song?
Among the Christmas selections I heard on the radio today was “Holiday” by the Bee Gees. Nothing to do with Christmas, except the title. 
Which come to think of it, for the larger share of the population of the USA and Western Europe at least, Christmas happening barely past the solstice makes what most Americans call Christmas Season not usually a “winter wonderland” for a majority of residents.
But this is a great point – Christmas is Nativity through Epiphany, but the modern popular culture in many places has flipped Christmas “season” to be the time before Christmas Day, actually Advent.
Prokoviev’s “Troika” from “Lieutenant Kije.” It’s become a bit of an Xmas thing, but it isn’t a Christmas sleigh-ride, just a sleigh-ride in general.
Still, it’s pretty!
There were a number of years when Three Dog Night’s “Joy to the World” was played on the radio only around Christmas time.