One problem, I think, is loose terminology, which causes confusion between longitudinal differences in populations (which will tend to be gradual along one lineage) and "lateral" differences - differences between co-existing populations/species, which, on the whole, are fairly categorical.
So modern birds are, in one sense, dinosaurs, in that they are dinosauriae i.e. descended from the same family as dinosaurs. But it would not be true to say that dinosaurs are birds, just as it is true to say that all cars are vehicles but not true to say all vehicles are cars. But the same problem would arise if someone designed a car and called it a Vehicle. Then it would be true to say that all Vehicles are cars, but still untrue to say all vehicles are cars.
So we should be more careful. If Fish=modern fish, we are not Fish. However if fish=the family of animals of which a fish-like animal was the common ancestor, then we are fish. But better to say we are all sarcopterygii, in that modern fish, modern birds, and modern humans, as well as extinct dinosaurs, descended from the ancient animals we have named sarcopterygii, and which looked a lot more like a modern fish than birds, humans or dinosaurs.
But birds, humans and dinosaurs did not, clearly, descend from modern fish.