I'm suggesting that people see dolphins sharing the (easily-observable) characteristics they do with tuna and aren't especially interested in the respiratory or reproductive aspects--particularly because the category of "air-breathing fish" is wrong in some sense by construction.

They are also wrong on the biochemical differences, but insofar as fish are pretty much just "all the animals in the sea" to the general public, I'd say the scientific and the popular use of the word probably diverge.

Perhaps I'm being too much of an economist, but I figure that the actual need for most people to know most of these biological details is limited, and the fuzziness in the popular use of the word arises from a low-effort sweep of the observables.