Fish is a polyphyletic group. It's a shorthand to refer to various lines of aquatic vertebrates with a similar anatomy. It's not a clade but that's not the only way to logically group organisms. People trot it out like a "gotcha" or just misuse it in much the same way they don't understand speciation (or most science terminology, to be frank)

We are not fish by anyone's honest definition, but "there's no such thing as a fish" is the kind of attention-grabbing false revelation I hate: it's the headline with none of the understanding to actually learn something.

(I'm not annoyed at you, I think you understand perfectly based on your wording)