The history of that particular style of two-color-two-strand knitting is more mysterious than you might think! It's commonly called Fair Isle technique, or stranded color knitting. The particular example you link is in the Scandinavian style of colorwork, which traditionally uses motifs like reindeer, geometric snowflakes, evergreen trees, etc. They weren't truly "Christmas sweaters" in intent though, just sweaters with wintery things on them. Scandinavian style motifs emerged somewhere between the late 19th and early 20th centuries. You can read quite a bit of this book before Google Books cuts you off, navigate to page 40 for some info on the development of this style of knitting. Fair Isle knitting motifs (from Scotland) tend to be smaller and more abstract, mostly different crosses, hearts, maybe as wild as a very abstract flower. Fair Isle color knitting is a little older, probably 19th century, but it got very popular outside of the Scottish Islands in the 1910s-1920s. There is a semi-famous painting of Prince Edward wearing one from this time.

The bigger multi color motifs (like HELLO it's a BIG SANTA FACE on your chest) are more modern, I don't start seeing patterns for that in American knitting books until the 40s, and the earliest example I have in my squirrelled away files is a book of patterns putting big cowboy things (hats, boots, horses) on children's sweaters. These use an entirely different knitting technique called intarsia, so they're not totally comparable. You juggle multiple colors on bobbins instead of only 2 colors at a time. I'm not very familiar with the history of intarsia though, because honestly I just don't like it! D:

The usual go-to book on the history of knitting is A History of Hand Knitting by Richard Rutt, but it's old, and Rutt is Extremely British about knitting history (read the first review!)