Primates, canids, felids, and ungulates are all subsets of mammals and all have further subsets. Mammalia is further a subset of animalia. When we’re discussing categorizations, it’s often helpful to have multiple levels of categories. I’m not sure why you seem to be calling out specificity as a gotcha, when my argument isn’t at all that we don’t need multiple terms. It’s that we should consider these things in terms of similarity and specific differences, not throw away a term as useless as the article and its headline suggest.