While fish is colloquial term, i have already given biological classification names that are not colloquial, in past posts:
Vertebrata (subphylum of Chordata)
Agnatha (infraphylum of Vertebrata) - “jawless fish”
Cyclostomi (clade) - hagfish, lamprey
Gnathostomata (infraphylum of Vertebrata) - “jawed fish”
Chondrichthyes (class) - “cartilaginous fish” - sharks, rays
Osteichthyes (superclass) - “bony fish”
Actinopterygii (class, or clade) - “ray-finned fish”
Sarcoptyerygii (class, or clade) - “lobe-finned fish”
Of the above list, the infraphylum Agnatha is considered paraphyletic. The rest are monophyletic.
Also, hagfish, or class Myxini, is listed under the Vertebrata, that based on their molecular data, but all living species of the hagfish don’t have vertebrae, because at some point in their evolutionary history they did have vertebrae, but then lost the vertebrae.
This is why some biologists, instead of using Vertebrata, they used the Craniata as the clade of Chordata, animals that have bony skulls. So the Myxini (hagfish) do fall under the Craniata.
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Note that while subphylum Vertebrata is monophyletic for all vertebrate animals, the term “invertebrate” is paraphyletic and umbrella term for all animals that do not have vertebral columns.