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.
* 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).
See also https://slatestarcodex.com/2014/11/21/the-categories-were-ma...
But none of them include marine mammals such as seals, dolphins and whales. And none of them include penguins, even though penguins flying through the water on their little wings are impressively graceful and fast. None of these animals are fish to me under any meaning of the word.