The dog's name is unknown, but he appears to be a Siberian Husky. He is possibly an early version of a K-9 police unit, which is likely because of Blackwater's police force being the most advanced in the game.
He accompanies Marston on Nightwatch assignments in Blackwater or he will otherwise just be running around Blackwater. If the player kills the dog with a gun or melee weapon, this will result in a wanted level. However, if the player charges at the dog at full speed with a powerful horse, the dog will die but the player does not become wanted and just loses 50 honor. If lassoed, he will bite and cause damage to the player.