Basic "Pay Attention!" message
OR - just simply a pet companion (though I think that would not fit really old decks, from the days when I think that wouldn't have been something "normal".)
True! In older decks the Fool seems to be showing us he's a fool not by heading off a cliff but by being harried by that dog. Any rational person would avoid a dog that was coming after him, nipping at his heels or getting a bite into his trousers. But the Fool doesn't even notice the animal, he's so distracted.
When I was a child, I daydreamed a lot--I think more than most, but it's hard to know--and some older teachers got very upset at this. I never quite understood why. Letting the mind wander seemed a very good thing for inventing, creating, etc.--and articles I later read backed this up. But I also read a lot of history later on, and what life was like back in the bad-old-days, well, it made a lot of sense for those teacher to be impatient with kids who weren't paying attention. Bad weather, wild animals, rough terrain, and, yes, other people trying to steal your stuff, all these old-time dangers, some of which these old teachers had gone through, needed a person's full attention. Or else, the person could very easily and quickly be dead.
This is still true today depending on what you're doing; but we're pretty lucky. We live in a time when we can stroll through a park and let our minds drift. No danger that, if we do so, we'll get lost in the woods, fall into a ravine, miss signs of a coming storm or get eaten by a wolf. So the dog, to me, is at it's most basic a throwback to this message. As in old fairytales, like Little Red Riding Hood, it's warning us to "Pay attention!" Don't be stupid. Don't be a fool and let your mind wander. Not because daydreaming is "bad" in and of itself, but because there are so many dangers out in the world. If you let yourself become distracted while on the road/out in the wide world, you're a fool. And fools don't end well.