This issue rests entirely upon how we define personhood: any entity that we recognize as a "person" is allowed all the rights associated with it. Centuries ago, dark-skinned peoples were not considered persons, and would not have been allowed copyrights.
Fast forward to this new millennium, and dark-skinned people can now establish copyrights, but Naruto the macaque couldn't establish his personhood and an associated right to own copyright; Naruto is still not a person. Courts have also ruled against those trying to establish personhood for other primates. Most recently great apes were demonstrated to pass a false-belief test, indicating they possess at least a rudimentary theory of mind, but I doubt even that will sway our hard-hearted xenophobic courts.
I'm not hopeful that a court will be willing to grant any AI the rights of personhood, including the right to establish copyrights, when even our closest genetic cousins with obvious rudimentary sentience can't pass muster.