Why is Fur Elise considered a Christmas song? Does it have anything to
do with Schroeder playing it in "A Charlie Brown Christmas"? Seems
like a lame reason. What about "My Favorite Things"?

"Für Elise" is certainly not considered a Christmas song in Germany,
so I expect that in the States the "Schroeder connection" may be the
reason.  Nice tune, maybe the only classical music some have heard,
and only heard there, so they link it to Xmas.  "Jingle Bells" isn't
related to Xmas either, but we all accept it as a seasonal song.
"My Favorite Things" from "The Sound of Music" also has nothing to do
with Xmas, but, of course, the title suggests presents, but only the
"Brown paper packages tied up with strings" could qualify.
Have both melodies been picked up for background music for Xmas shopping?
Their both being "better" musically, could account for their winning acceptance.