Can't adapt that argument to fish, because descendants of the ancestor we share with fish are not generally called fish, the way descendants of the ancestor we share with orangutans are generally called apes.

I'm not sure this holds water: a common-ancestry approach would have to take in lobe-finned fishes like the lungfish, who're more closely related to tetrapods but are called fish on the basis of a morphological similarity derived from a common ancestor. Essentially the same process as for apes. They're in good company, though: there are plenty of traditional taxonomical groups which turn out to be polyphyletic when you take a cladistic approach, including reptiles.