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)

From what I understand, this is sorta like a hangover from pre-DNA taxonomy. We went "yeah, those all look like fish, we'll put them in the fish group", only to find out later that a bunch of them weren't very closely related at all. So now we have a 'fish' group that's a total mess, and we're in the middle of getting it organized and re-labelled.

Can't we just un-fish it like we do for other clades when we need to?

"There's an ape in the office!"

"Yes, his name is Tom. More importantly, he is a human being, and we don't refer to them as apes outside of an academic context and even then, only when necessary."

[Tom eats a banana, screams at an intern, and starts picking his nose]