I searched for the origin of this term and found that it was the title of first Stan Laurel and Oliver Hardy movie. Stan is homeless in the street and is befriended by a stray dog. There are a series of events including the dog making friends with a poodle, a dog show, a jealous boyfriend, etc. Wikipedia says Laurel and Hardy play independently of one another and are “…not the comedic team they would later become.” Of course calling someone a “lucky dog” means that you believe they are very fortunate.