The 'Les Miserables' star, 31, adopted the mixed breed last month and renamed him after one of the most famous characters in pop culture, reported Us magazine.
Kenobi joins the Brooklyn-based couple's chocolate labrador Esmeralda, who the actress has had for years.
Hathaway, who wed actor and jewellery designer Shulman, 33, in 2012, is next set to star in Christopher Nolan's star-packed sci-fi drama 'Intersteller'.
She's currently filming the Nancy Meyers comedy 'The Intern' with Robert De Niro.