By Schrödinger's cat, we don't actually mean a feline animal. Schrödinger's cat is a metaphor inspired by Erwin Schrödinger's famous thought experiment, which says that if you put a cat in a box with a vial of poison that could release at random, you don't know whether the poison has released and killed the cat until you open the box. As a result, as long as the box is shut, the cat is both alive and dead.
This phenomenon of being in two states at once is called superposition, and it's a fundamental property of quantum physics. Just as Schrödinger's cat occupies both states of "alive" and "dead" until you open the box, quantum particles occupy multiple states until they're measured. That's what we mean when we say that physicists are "breeding" Schrödinger's cat: they're turning one particle in superposition into two particles in superposition that are linked together, or "entangled," which physicists did successfully in 2016. In the real world, "breeding" Schrödinger's cat could be an important step in developing better quantum computers.