I recently remembered this movie from 1991 (some sources say 1995, the backstory of this movie seems really all over the place) called The Seventh Brother. I watched it as a kid in Finnish so I don't know the characters names in English, but to tl;dr the movie, it's about a dog who gets lost in a forest and is then rescued and adopted by a bunny family.

I own a VHS of the movie, but it's somewhere so deep in my closets I haven't had the chance to rewatch it yet. In that version, the dog is abandoned by his owner and is thrown down a cliff out of a car window. In the version I saw online, he chases a frog and falls down instead. It has all the same climbing scenes too, just a couple of the scenes are off and deliver a whole different beginning for the movie.
Another difference is that the owl is telling the "story", but in the version that I saw most of the scenes with the owl are gone and replaced with a woman explaining the story while scenery is shown.

I didn't find much about this online and I was hoping someone would know something about it, I know it's a stretch but doesn't hurt to try right?