There's not an obsession with robotic animals in the book, it's an obsession with animals and keeping them as part of the new religion, Mercerism, and its hyper-focus on empathy, and consequently displays of empathy. Caring for an animal is part of that. The robotic animals are not meant to be recognized as robotic, they're surrogates so people can still display their care of an animal and demonstrate empathy even if they can't afford a real one. Real animals being rarer and more expensive, and of course dying much more easily than a machine, as happened to Deckard's prior to the events of the novel.