It's true that there is a group of animals known to most people as "fish", and that naming serves a valid purpose, but you said that a "strict cladistic interpretation" would say that humans are fish, which is not true. Cladistics would say that humans and "fish" are vertebrates, that humans and most "fish" are jawed vertebrates, and that humans and a smaller group of "fish" are teleostomes. Agnathan, Chondrichthyan, and Osteichthyan fish are all monophyletic groups, none of which includes mammals. This is different from the "birds are dinosaurs" statement, which is true because Aves is a monophyletic clade nested within another monophyletic clade (Dinosauria).
You also said that that method would be imprecise, which is also not true. The hierarchical nature of phylogenetic classification (with clades nested within clades) allows for explicit hypotheses of evolutionary relationships. For example, Hominoidea = Hominidae+Hylobates (gibbons); Hominidae = Homininae+Pongo (orangutan); etc.