I think you are conflating legal rights with moral rights.

It is largely, although not entirely true, that nonhuman animals don't have legal rights (although this is slowly changing with the recognition of personhood for nonhumans in several important legal cases). However, it is also true that African slaves in America, for example, didn't used to have any legal rights. Does that mean those people had no moral rights? Perhaps you think so, but I would disagree - I believe they had an inherent right to freedom, self-determination, life, respect, etc.

By the same token, I believe nonhuman animals also have inherent rights, whether or not the law (whatever that means) currently recognises those rights.