NEW YORK—One recent morning, Reilly Richardson woke up to being watched. She jumped out of bed, went to the window of her Manhattan apartment and fell to the floor when she saw the peeper. “It absolutely scared the you-know-what out of me,” she recalls about the 2-foot-tall owl staring back at her. “It’s New York City. It’s the last thing you expect to see.” Richardson, 31, and an actor, didn’t know it then, but her feathered visitor was Flaco, an orange-eyed Eurasian eagle-owl who became a New York celebrity after he escaped a zoo in February and turned leafy Central...