I agree with etiolate although I'd add that they also preserved the ice-cynicism metaphor. You can withdraw from society and wallow in isolation instead of choosing to come to terms with the way the world is, and worse, when you lose the ability to see the goodness in others it has an effect that extends beyond you (Elsa causes a winter and freezes Anna's heart). And the warmth of love restores order, etc. But it really is far removed from the original fairy tale. And Hans has no meaningful place in it at all. I guess he's there because you don't need romance to fulfill your life, you can find love among your family and friends, and while that's a good message it's still a much more modern tale than the original fairy tale. In the Snow Queen, Gerda awakens Kai with a kiss, and it's love (even if it's childish love), and when they return to the village they're adult lovers.
I think the original Little Mermaid is better than the Disney movie, too. Even though she doesn't end up with the Prince, love allows her to transcend into a heavenly spirit, which reminds me of an obvious Shakespeare quote. The point of love is not that you obtain a lover but that you are awakened spiritually. And that's a really powerful idea.
So tying it back I like Peterson's ultimate sentiment: dispense with old fairy tales at your own risk. There's thousands of years of incremental knowledge in there. Trying to recreate it is, demonstrably, a hard task. But, that said, it's still funny seeing him go on and on about some dumb movie.