The text for the Dog Companion is as follows:
Take a small monster token to represent the Dog. Put it in your room. (Use a token of a different color from other monsters, if any.)
Once during your turn, the Dog can move to any explored room up to 6 spaces away, using doors and stairs, and then return. It can pick up, carry, and/or drop 1 item before it returns.
The Dog isn't slowed by opponents. It can't use one-way passages or rooms that require a roll. It can't carry items that slow movement.
So in other words, the instructions say:
- Use a token to represent the dog
- The dog can go to another room (and can carry items, perhaps to trade with another investigator) and then come right back
- The dog can't get stuck by using a one-way door or a room that requires a roll
Since the dog always comes back to you immediately, and there is no situation where a dog can go somewhere they can't return from, what's the point of having a token for the dog?