I have never heard anyone say THAT, but I’ve heard animal rights activists say that their rights or any human’s rights, are no more important than that of a slug or amoeba,
Animals’ rights include their right to exist and to exist in suitable habitat.
Bears think they have a right to maul or kill you for a sandwich, or if you don’t have a sandwich, But bears are wrong. Covertly, animals don’t get a trial if they attack or kill someone.
Animal rights activists may push for veganism, but I think we have a right to eat animals, as all carnivors eat other species. But bears and big cats don’t have rights to eat us, the dominate species.
If Animal Righrs Activists think our rights are equal to animals, they usually mean we have no rights over their habitat for human development, like office parks and subdivisions, but will usually compromise with making spaces we take over more environmentally friendly, even for the animals.