The idea is that birds are what they descended from and therefore are dinosaurs. So then dinosaurs too are whatever they descended from.
Surprise ... birds are dinosaurs and dinosaurs are what? If we are clever and accurate to say that birds are dinosaurs then what should we say that dinosaurs are in order to use that same logic with consistency?
A clade is just a group and all it's descendants. So if two things are members of a clade, then so are their common ancestors. If Chimpanzees and Gorillas are both apes, then so are the common ancestor of Chimps and Gorillas. And so, necessarily, are any other descendants of that common ancestor (including humans). So either humans are apes or "apes" isn't a word that describes a clade, it's describing something else.
For instance Fish clearly doesn't describe a clade, because if it does then it includes whales (and humans), as both are more closely related to some fish than those fish are to other fish.
From an evolutionary perspective clades are a pretty useful classification system, because they're based on relatedness. If dinosaurs are a clade, then birds are dinosaurs. But there are other classifications that can be useful other than cladistic ones. I think fish is still a useful category, even if it doesn't include humans. It's useful because animals that adapted to life in the ocean over deep time have some shared characteristics, and perhaps more importantly life that moved on to land changed in meaningful ways such that a classification of "fish that stayed in the ocean" actually does capture something important.
But is dinosaurs like that? I'm not sure. Birds are a clade of their own, it's entirely possible to talk about birds as birds (one branch of the dinosaur family). Generally I think a cladistic approach makes the most sense here. Birds are dinosaurs, even though not all dinosaurs are* birds.
*Probably should be "were", since at the moment all dinosaurs are birds.
