There is also a guinea pig, some other birds, lizards and even a few human beings. But this is mostly a dog story, full of capers and complications arising from the arrival of Duke (the big shaggy one voiced by Mr. Stonestreet) into the household Max shares with Katie (Ellie Kemper), his millennial owner. There is rivalry and resentment and eventually friendship. “The Secret Life of Pets,” written by Cinco Paul, Ken Daurio and Brian Lynch and directed by Yarrow Cheney and Chris Renaud, is like one of those picture books about how to deal with a new baby, but with talking animals.