Many remember the scene where Professor Hinkle is told by Santa that he has to write an apology "a hundred zillion times" to receive any goods, but the dialogue originally began as a threat that if he touches the hat he'll never receive another Christmas present again. Hinkle then names examples of things he would want. If there was continuity to the discussion, after he said that he had to make a living too, Santa would have said something like "Well you'd better not touch that hat, then".
The order to write the apology could have been a test to see how repentant Professor Hinkle was, or how seriously he took Santa's words. Even if he didn't write that many, if Santa looked in and saw him doing the job, it would mean Hinkle took what he said to heart.
Besides, real life conversations don't always have perfect continuity.
In Rudolph and Frosty's Christmas in July, Frosty and Crystal have children! Um...how do snow people procreate? Did they make them and bring them to life? Magic? Um...
That seems likely, as that's how Crystal and the snow parson who married them were created.
As this review says, "Thumpity thump thump, thumpity thump don't ask how Frosty mates."
Why does Frosty always say "Happy Birthday" when he gains consciousness? Whose birthday does he think it is? He can't be referring to himself, you don't say "happy birthday" when it's your own birthday!
He thinks it's someone else's birthday (for some reason)....
Well, it's Christmas and he's made out of Christmas snow. It's not hard to figure out [[Useful Notes/Jesus who]] he'd want to wish a happy birthday...
How is it Crystal immediately knows Frosty's name when she first comes to life?
The ticket booth operator lists a bunch of locations that would be counterintuitive to travel to (Saskatchewan is in central Canada, Hudson Bay eastern Canada, Nome is in FAR western Alaska, a whole trip across the continent westward, the Klondike is in the Yukon, a southwest backtrack) until you realize he was humoring the kids and trying to say that there's no way to travel by train to the north pole, which is a bunch of ice floating in the Arctic ocean. There isn't even a land bridge between Canada's Ellesmere Island and Greenland. Had the kids had the $3000 for the ticket, he would have had to admit he was making stuff up or make up a story about train service not being in for one of the lengths of the journey.