if it can only breed with other cats it is a cat and not a dog.

By this overly simplistic definition you would count infertile cats as non-cats. You also completely lose the ability to differentiate the species of non-sexually reproducing organisms. And then there's the insanity that is hybridization. Is a mule a horse or a donkey? They can occasionally reproduce with either of them.

In reality, differentiating species is a complicated science/art that involves not only the reproductive isolation, but also morphology and genetics, all with the goals of coming up with the most useful definitions for species.

In just the same way with economic systems. Workers in different societies and periods exercise different levels of control (which is the underlying meaning of ownership) on the means of production, it's not black and white. It can be very roughly defined as (workers control over the state) * (relative assets of state-controlled enterprises) + (relative assets of co-ops or other directly worker-owned enterprises). There is of course a lot more nuances to be discussed, such as the exact distribution of control between different subclasses of workers, or the social hierarchies arising from the structures of control. Once again, the goal is to come up with a useful scale to gauge how much a certain country or region has progressed on its way to socialism, to learn from their mistakes and to build better governance systems in the future.