Alternative Character Interpretation: Was the Postman crying at the end because he was defeated or were they Tears of Joy? If you look below, one of his reasons for hating Christmas was being put on the naughty list, so it's possible that he was touched that the penguins gave him a gift, even if it was raw fish.
Fridge Logic: The Mondegreen which starts Olive on her journey was changed in the adaptation likely for this reason: It's a major Contrived Coincidence everyone Olive heard singing happened to be on that particular line and seemingly stopped at that point, or she would have realized it wasn't about her.
Jerkass Woobie: The Postman. The reason he hates the Christmas season is because it greatly adds to his workload. People who have to consistently work during Christmas can probably sympathize with him. Although, it is implied that the true reason is because he's mad at Santa for putting him on the naughty list.
Memetic Mutation: "Bah, Bug and Hum" is modestly popular enough to spawn a few Fan Vids featuring villains from other properties.
Moe: Olive manages to be one and a strong character at the same time.
Strawman Has a Point: While the Postman is already a Jerkass whp went too far by trying to ruin Christmas for everyone, his dislike for the holiday still holds water. Since Christmas seasons greatly adds to his workload, but we the audience are supposed to root for Olive.
Tear Jerker: The scene where Olive thinks that Tim is getting a new dog and replacing her (unbeknownst to her, he was apologizing for snapping at her).
The Woobie: Olive wants to save Christmas despite people, the weather, and bad luck impeding her quest. Viewers can't help but feel sorry for her when she almost feels like giving up.