Genus, specie, thylum...

[Undertow]

This falls into the age old question 'How do merfolk reproduce' but thats not the question.
You see in the writing of my novel Undertow, I seriously kicked around the idea of merfolk being a member of the monotreme family (like the echidna and platypus). My reasoning behind this was they laid their young in eggs like most sharks (a couple of species give birth to live young) but they suckled their young (hence the rack hahha). I would up abandoning it and having them as simple warm blooded mammals which is easier as they come to surface to give birth, they need sunlight to regulate their body temperature and they reproduce like most mammals.
I dont know....thats just what I did for my story but what are your thoughts?

[Princess Kae-Leah]

In my series, merfolk are purely magical creatures and as such, they are classified as neither fish nor mammals. They breath water like fish(they have gills on the back of their necks) and reproduce somewhat similarly to mammals, but not quite(they basically intertwine their tails and reproduce by their souls merging, they don't have private parts the same way mammals have). They are also vegetarians and can live as long as a thousand years. Essentially, merfolk, fairies, centaurs, and other magical/mythical creatures in my series are a middle ground between being natural beings and being angelic, divine beings.

[malinghi]

I agree that you shouldn't get to hung up on where they are in the animal kingdom, cause ultimately they have traits that are mutually exclusive between fish and mammals. There isn't anywhere you could put them where having breasts and breathing underwater makes sense.

[Joy&RaptorsUnrestrained!]

By the way, I think you mean "Phylum," not Thylum.

Mermaids as monotremes is an interesting idea, but I'm not sure it (forgive the pun) holds any water, given how different monotremes look from classical images of mermaids. Platypi are able to dwell on land and in water, which makes your theory especially intriguing, but they don't share the general shape or size of merfolk.

It might be best to consider them part of a class of mythical hybrid creatures, with different orders for mammal/mammal mixes (such as Satyrs or Centaurs), mammal/birds (Simurgh, Harpies, Pegasi, Griffins), mammal/reptiles (Naga, Ophiotaurus, Lizardmen, Kappa, Gorgons, etc), mammal/fish, fish/reptiles, reptile/birds, mammal/insects, mammal/arachnids and so forth and so on.

In Courtney's Underwater Tales rpg, I used the scientific name "Sirenium Lusignan Atlans" (genus, species, and sub-species) for my character, a "Mid-Atlantic Melusine," with Sirenium being the genus for mers in general, Lusignan/Melusine referring to mers with flying-fish wing-fins (based on the story that Melusine, upon learning that her husband had disobeyed his promise not to spy on her as she bathed in her true form, leaped from the window and flew away, and would return to fly over the castle when one of her sons died), and Atlans for the subspecies found in the Atlantic. Looking at the scientific taxonomy, I've made up the following...

Life, Domain (Eukarya), Kingdom (Animalia), Phylum (Chordata, subphylum Vertebrata), Class (Fabulosa), Order (Mammalichthyes), Family (Syreni), Genus (Sirenium), Species (Lusignan), Subspecies (atlans)

Mermaids are not single-celled organisms, or bacteria, so they must be in the Eukarya domain, which includes the Animal Kingdom, which includes the Chordata Phylum (which includes vertebrates and some related invertebrates) since we assume that mermaids, as mixes between fish and humans, have backbones. Then we add the "Mythical creature" class, which I took from the Latin word for mythical, Fabulosa. For orders, I combined the words Mammalia and chrondichthyes/osteichthyes as Mammalichthyes, to represent Mammal/Fish mixes. I chose the family Syreni to define creatures with humanlike upper bodies and fishlike lower bodies, which would also include tritons and ichthyocentaurs, but not fish-men like creatures from the black lagoon. Sirenium narrows the field down to merfolk, and Lusignan for flying fish mermaids, and Atlans for the Atlantic sub-species. Other species names might include Ningyo (for mers who are more fishlike than humanoid, even above the waist), Nixe (for freshwater mers) Havmennesker (for mers resembling those in the Little Mermaid, name based on the original title "Den Lille Havfrue" and examination of the words "merman" and "people" in Danish), and so forth, with appropriate regional or physical feature (coloration, fin shapes, etc) subspecies names.

[green52]

I like all the taxonomies presented. Malingi makes a good point that it won't really be possible to explain an evolution which is clearly so fish-like and so human, but if you're willing to suspend disbelief, monotremes, porpouises or ceateans, or a lineage that includes mythical hybrids all are convenient options.

[Undertow]

Nah I realize that you cant categorize them (though I do like the above explanations) and I didnt I just said they were mammals and left it at that hahha because I didnt want some nerds chasing me down the street with a textbook xD