That argument is not very cogent because the idea of owning a person is too outrageous.

But if you change it sligthly to owning copyright over some dog breed, that emotional objection would dissipate in my opinion.

So if someone were to cross two or three breeds of dogs and make a stable, new breed, can that someone patent it and own it?

What about every single pup that is eventually born? Can the breeder claim intellectual property rights over it?

Of course this is nonsense, but some companies will try and are trying their luck.