No one argues that nonhumans should have HUMAN rights. But across time and space, it is not uncommon to grant legal personhood and thus legal rights to nonhuman entities such as corporations, ships, rivers, animals, or religious idols. There are different reasons for doing so, ranging from instances in which extending legal personhood helps resolve human conflicts (ie. corporations) or reflects Eastern or Indigenous ideas about how humans relate to nonhumans (ie. rivers and religious idols).
Further, rights are not a zero-sum game. The idea that affording rights to one group detracts from our ability to protect another has no empirical basis. The reality is that humans have an innate tendency to anthropomorphize and roboticists are only going to continue developing robots with maximal human likeness. The only question is, how should we incorporate such entities into our moral and legal universes?
I develop these ideas in greater detail in my new open access book, Rights for Robots: Artificial Intelligence, Animal and Environmental Law (Routledge 2020), available for free here: https://www.taylorfrancis.com/books/rights-robots-joshua-gellers/10.4324/9780429288159.