The answers by u/hillsonghoods and u/phosphenes are excellent introductions to canon formation in general and musical canon formation in particular, but I would like to add a few more factors that I think are extremely pertinent for the particular formulation of the American Christmas music canon:
- The extreme, record-breaking success of Bing Crosby's "White Christmas" in 1942. It's hard to overstate how well this record did: It's not only, by far, the most successful song in the Great American Songbook, but the biggest-selling single of all time, clocking in at an astonishing 50 millions singles sold and never beaten despite the record industry still having some forty years of constant growth ahead of it. Sure, Tin Pan Alley did have a history of successful Christmas songs before "White Christmas", like 1937's "Santa Claus is Coming To Town", but the absolute sales tsunami of Irving Berlin's song produced a host of copycats. Much of what we think of as the Christmas pop music canon was written in the subsequent decade, from "Have Yourself a Merry Little Christmas" to "Rudolph, the Red-Nosed Reindeer". "White Christmas" was a touching point throughout. In his autobiography, Tin Pan Alley great Sammy Cahn memorably indicates the sheer reverence that the song was held in:
One day during a very hot spell in Los Angeles the phone rang and it was Jule Styne to say, "Frank wants a Christmas song." Most Christmas songs, I should say, are written in the heat of June or at the latest July, in order to give the singer, publisher, the record company, the promotion people, and the weather a chance to get together. "Jule, we're not going to write any Christmas song," I said. "After Irving Berlin's 'White Christmas'? The idea's just ridiculous." Jule said: "Frank wants a Christmas song."
- The advance of recording technology and the hi-fi boom. High-fidelity recording, the kind we're used to today, didn't really reach its full height until the second half of the 1950s. This period also saw "Hi-Fi", high fidelity, become a significant marketing factor by record companies and record player manufacturers. The particular style we associate with Christmas songs of the era – soft crooner singing recorded by highly-sensitive microphones, full orchestras with large string sections, and so on – was simply not possible to achieve earlier, and earlier material suffers in comparison. As a result, much of what we think of as standard recordings of Christmas music is actually re-recordings made using this improved technology. Take "The Christmas Song" by Nat King Cole, where the (second) 1946 recording was the first hit version, but where a 1961 re-recording is almost universally the version ending up on compilations today.
- The rise of the 12-inch LP album as a significant commercial format in the first half of the 1950s. Albums didn't start outselling singles until a decade and a half later, but there was still a significant market for them, and Christmas albums, even more so than Christmas singles, were very successful in this period. (As a comparison, by 1960, three Christmas singles had reached #1 in the pop charts, besides "White Christmas" also Jimmy Boyd's "I Saw Mommy Kissing Santa Claus" and "The Chipmunk Song". In the same period, six different Christmas albums had reached #1 in the album charts.) The rise of the Christmas album as a format is very important for canon formation! While many hit singles would receive a handful of recordings of the same material in subsequent years, it was only with the advent of the album that you really get an explosion of new versions of the same song. The few years in the early 1950s helped cement what songs formed part of the Christmas music canon partially because this was when a lot of artists had to pick what to include on their Christmas albums from the previous decade of hits, and some songs became almost universal inclusions, while others remained in the dust. Your Frosty and your Rudolph got to stay onboard, while other somewhat successful novelty singles like "Angie, The Christmas Angel" or "Suzy Snowflake" got left out in the cold, and never really recovered.
- Finally, the stylistic changes of the 1950s are a significant factor in why Christmas music canon formation stopped in the late 50s. The market shifted away from traditional pop novelties to a large extent, and (at least for a while) from the major labels and their giant, orchestrated recordings to independent labels, younger musicians and smaller recording studios. (It is notable as discussed in my previous answer here that the enduring Christmas songs that supposedly are in this "new style" of youth music are still part of the Tin Pan Alley/major-label machine, and only had limited success on first release.) Tin Pan Alley retreated, and kept working on older styles instead. Many successful recordings of Christmas music would continue to be produced, not least albums, but most of them were now consciously targetting an older audience (enshrined as the Easy Listening Billboard chart in 1961) and evoking nostalgia of the "White Christmas"-related pop of two decades earlier, or rerecording the same material, severed from contemporary youth rock. No wonder new songs were hardly added to the canon!
Sources:
Cahn, Sammy, I Should Care: The Sammy Cahn Story. New York: Arbor House 1974
Vinylmint history of the record industry