Such a category is called paraphyletic. It can be informationally useful if the excluded subgroup is far-divergent from the overarching group, such that it has gained characteristics not shared by the others, and lost characteristics otherwise shared. But the less divergence has taken place, the harder it is to justify a paraphyletic category. The category “reptile” (excluding birds) makes sense today, but it wouldn’t have made sense in the Jurassic period. The mammal/cetacean distinction is somewhere in the middle.
Animal/human is different because the evolutionary divergence is so recent that it’s difficult to justify the paraphyletic usage on biological grounds. Rather this is more of an ingroup/outgroup distinction, along the lines of βαρβαρος (“anybody who isn’t Greek”). If humans learned to communicate with e.g. crows, the shared language probably wouldn’t have a compact word for “non-human animal,” although it might have one for “non-human non-crow animal.”