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).
But actually I think coelacanths are quite a fun example. I hadn't heard of these before, thank you!
Yes, they're not tetrapods, but (I've just discovered) they're not even vertebrates (no spine). According to my definition, they shouldn't be fish, but they do seem quite fish like.
They are chordates (they have a spinal cord, just no backbone for it), so I could expand my definition to any chordate that isn't a tetrapod. But there are some rather non-fishy chordates [1] so that doesn't work either.
[1] https://en.m.wikipedia.org/wiki/Tunicate
(For those that don't know, the top level subclassification of animals is phylum. There are a lot of phyla but a common ones are chordates (all vertebrates plus a few odd animals like discussed above), arthropods (insects and insect-like things like spiders and crabs), and molluscs (like slugs and clams). When I was at school, animals were just vertebrates or invertebrates but the reality is more interesting. I went down that rabbit hole when I found out that, weirdly, octopuses are molluscs.)
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.