Some biological taxa are clades: a clade is a taxon that contains a descendant of every included organism. For instance Mammalia is a clade, while Reptilia is not, since birds aren't reptiles but are descendants of reptiles. There are biologists that wish that we used a phylogenetic classification scheme, one where all taxa are clades. But that's not what is traditionally done. Let's consider the hypothesis that the traditional approach is right in the sense that it cuts nature at joints. Then a taxon can change from being a clade to being a non-clade. I assume that Reptilia changed in this way when birds evolved. And whether such a change has occurred is a substantive question.
In principle, then, it is possible for the kingdom Animalia, which I understand is normally taken to be a clade, to change into a non-clade. And it is a substantive question, then, whether such a change occurred when humans evolved. It could be the case that sapience marks such a departure that we are a new kingdom, and Animalia is no longer a clade. I think a close relative of this thought--albeit without evolutionary connections--is behind the ordinary person's (as opposed to a philosopher's) resistance to the idea that we are animals: personhood is such a transformative feature that it marks a completely new kind of organism.
But while the question is substantive, it's not tenable to say we aren't animals. If we are not animals, it seems we aren't mammals. (Maybe more can be said, though?) But if we aren't mammals, then various natural kind-based explanations fail: we can't say, for instance, that we have complex bones in the inner ear because we have mammals.
Note, too, that the question raised in this post is orthogonal to the question that animalists are concerned with. For all that we animalists need for our positive theory is that we are organisms--whether the particular kind of organism we are is an animal, a plant, a fungus or something else is not very important for the theory.