Eels are fish

Apparently we are all fish. Or fish don't exist.

To explain: if you want to define a taxonomy in which all things that look like fish and swim are 'fish' then we are too. We are more closely related to most 'fish' than sharks are. I.e the last common ancestor of herring AND sharks is older than our & herring's LCA.

Looking at the Wikipedia article for fish, it looks like a reasonable definition would be:

* Everything in the subphylum vertebrata (i.e. vertibrates)

* Except tetrapoda (tetrapods: amphibians, reptiles, mammals and the like).

It's not perfect because tetrapoda does fit within vertebrata in a biological / genetic sense (as a sibling comment put it: fish is not a monophyletic group). But it's a precise enough definition that I don't think we need to claim that we're all fish or that there's no such thing as a fish (as the QI elves would say).

Mammals include orcas and whales

And orcas and whales are not fish.

Whales are fish that spout and have horizontal tail fins. (Currently re-reading Moby-Dick and that's the definition Ishmael comes up with.)

I think it's ok for there to be two meanings of "fish": a biologically formal (but not perfect) definition like I gave in my comment above, and a more informal meaning of "animal thing in the sea" that includes whales and even "starfish". It's very common for words to face more than one meaning. But that doesn't mean you can invalidate one by referring to the other.