So much is focussed on the ownership question: Do people own animals?
But here is another question: Do people own their children?
The law grants a child's parents unparalleled control over the life, and even the death of the child. A parent can determine when and what a child eats, whether they are clothed or not, when or if they can play, what friends they can have, and in case of medical decisions, when to no longer allow medical intervention to prolong their lives. They are potty trained just as dogs are housebroken.
In much the same way we make decisions affecting the lives animals that live with us, so to do we do the same with our children. So are they property? They don't have first amendment rights, since parents can easily, and legally, curtail any speech that they might make, they certainly don't have religious freedoms, as parents will always push their beliefs on their children as a way to teach them right from wrong.
In fact, more startling, a fetus is better regarded and protected than a child that is born. For once a child is born, their right to life is secondary to a parents' right to raise that child how they see fit, within the boundries of the law.
I submit that some people treat the animals living in their home better than they treat their own children. Now that is frightening.