Alternative Character Interpretation: Santa is in fact a big jerk in the original film because he never even "liked" Rudolph (if you can call it that) until he realized that he could exploit his nose. In other words, Santa only cares about Rudolph's nose, not Rudolph.
Rather than being an "abusive" parent, it could be argued that Donner, having grown up in that environment, knew the pain and suffering in store for Rudolph and wanted to spare him.
Did Comet exclude Rudolph from the reindeer games out of prejudice, or was it for the good of the other young reindeer who might be distracted and he chose the wrong words in Rudolph's presence?
Accidental Aesop: Rudolph's Shiny New Year has one about embracing the things that make you unique, but it sort of blurs into "if your physical deformity makes people laugh, let them. Be glad you're making them so happy".
Remember kids: Your differences will always be punished unless they can be exploited.
The Toy Taker aka. Mr Cuddles seems to be considered one of the better elements of the 2001 sequel.
Fair for Its Day: Seeing Rudolph find acceptance AFTER the others discover that his nose could be useful can leave a modern viewer with a bad taste in their mouth. However, in the 1960s, The Hays Code was still in affect, meaning that filmmakers could not show non-conforming characters in a positive light. At the time, it was radical for a mainstream family film to suggest that differences should not only be accepted, but celebrated.
Family-Unfriendly Aesop: No one (except your mother, apparently) will love you for your uniqueness, unless you're useful to them. Rudolph would still be an outcast and not a celebrity, if his defining trait could serve others no purpose.
Fanon Discontinuity: The sequel, Rudolph's Shiny New Year, suddenly has a completely different Rudolph puppet, despite starring the same character and taking place immediately after this special.
Not to mention he's also young again in all the sequels.
Or did he get his antlers trimmed once in a while?
Fans of both the traditionally animated 1998 movie and the "originals" will agree that the computer animated Rudolph the Red-Nosed Reindeer & the Island of Misfit Toys never should have happened. Arguably a song or two and using the original designs the other sequels altered are the only saving graces, if even that.
Expectedly every featurette of Rudolph follows the original story and song's formula of Rudolph being isolated and bullied for his nose (usually to the point of tears) until Santa employs him. The Rankin-Bass special really ups Rudolph's abuse however.
The Island of Misfit Toys. The concept of it is surprisingly depressing.
When Yukon is thought to be dead.
There's Always Tomorrow.
From The Island of Misfit Toys, Mr. Cuddles, after being exposed as the Toy Taker, singing his backstory.
The doll's line "I haven't any dreams left to dream."
Rudolph crying himself to sleep in the 1944 short.