I like how tragedy emerges here from explicit limitations: the dog can't read the signs and the graphical fidelity is low enough that the player can't either, so you're acting on incomplete information. A codependent relationship between dog and human shattered before the game even begins. Although this structure only emerges through multiple plays. There's not really time enough to discover that there are multiple paths out, the player will probably just pick the first exit she arrives at. The text on the sign at the end could use some visual contrast: black on brown doesn't really stick out for such an important element of the game.
Totally missing some sad dog voices. If you wanted to add an interaction, it could maybe make the dog whimper/whine. Or
cry. Or maybe the dog can try slowly dragging his owner away, but automatically drop him at a certain blizzard intensity?
I like the idea of footprints. Even if the dog doesn't leave them dynamically, you can still use them for a bit of worldbuilding by showing the owner's last few steps fading away at the start.
Right now, the characters are pretty flat, archetypal, defined mostly by the situation they're in. I don't think this
needs dialogue or a narrator, but whatever you can place in the world to give the owner more character could help. Was he lost? Could he have dropped a map, that's now fluttering around the clearing? Could he have been trying to get a GPS signal? Was he trying to accomplish a specific objective in the woods (hunting, building a fire, delivering a package) that might have left a trace on the world?