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The mermaids are human down to the knees. Their thighs are fused together as in sirenomelia, to form a single structure. However below the knees they have a fish-like tail with vertebrae and the other usual anatomy of a tail
What is the most vertebrate way that such a structure could develop in an embryo?
What is the most vertebrate way that such a structure could develop in an embryo?
Two ways spring to mind... neither are necessarily easily or reliably replicable. They are teratomas (content warning: real life body horror) and a kind of conjoined twin or chimera where part of the body of an almost entirely absorbed twin forms the additional spine.
It is possible that with enough genetic manipulation and mutation you might be able to get this to arise spontaneously, but I seriously doubt it.
However below the knees they have a fish-like tail
There's no reason that a fish like tail has to have a fish-like bone structure supporting it. Fused lower legs and feet could quite reasonably evolve into a tail with a cetacean-like appearance (eg. horizontal orientation whilst swimming, not vertical). You can use modern monofin swimming as an example of how this might work:
(image credit Aquamancer on deviant art)
With a genetic rather than prosthetic origin for the fin, the ankle flex is likely to be quite different and the toe bones will be greatly elongated, etc etc, but you get the idea. The shape and swimming style will be somewhat different to a fish or cetacean, but the basic appearance and style would work for your needs, I suspect.
edit: one possible solution:
Consider the possibility that your merpeeps do, in fact, have a cetacean-style tail which is formed at the end of a continuation of their spine like a regular vertebrate tail. Their legs are both fused together and fused into the tail. The legs are vestigial, and their muscles are unlikely to contribute to swimming very much if at all.
One thing you might lose here is the ability for your merpeeps to sit up on a hard surface out of the water, because resting your whole weight on a bent over bit of your spine sounds uncomfortable. The image of beautiful mermaids combing their long hair and singing songs to lure sailors onto their rocks whilst beached on their bellies like a harbour seal is an entertaining one, though.
Only artificially
I can't imagine any natural pressures that would make this happen. In fact you're more likely to have them either:
(A): Lose the legs entirely, elongate the spine, and have fins form by spines of keratin(hair) coming out of the elongated body and having webbed skin between them. Think aquatic lamia/snakeperson, or more aptly for mammals like us, some sort of dolphin-like lower body, though like the dolphin you'll find that we may become more streamlined as well, probably losing much of what we'd call a humanoid body shape for our upper, or in this case frontal, body.
OR
(B): Have the feet become what many swimmers put on their feet anyway. Fins. The bones in the feet would become extended and turn into cartilage instead of purely hard bone, while the extended toes would get webbing, creating the fins that an underwater humanoid would make use of but suffice to say this would make it so that they can only traverse on land on their hands and knees, assuming the hands haven't turned into fins as well at which point they'll probably need to use their elbows.
What you're seeking can only be done with functional-purpose surgery, where the body is artificially altered to serve a purpose, but this seems excessive compared to simply slapping on a pair of synthetic fins.
As per your example disorder, sirenomelia does do basically more or less what you're envisioning, with regards to the fusing of the legs specifically, but it is wholly detrimental and I severely doubt any kind of functional species could come from it unless, once again, the condition is altered artificially to make it less detrimental and more functional. [Sirenomelia][1]
Actually, now that I've thought about it...
It is somewhat possible for your mermaids specifically to come about naturally, but it is a rather convoluted way to go about doing this. It is a three stage process, laid out as so:
Stage 1: Humans regain their tail. I don't what kind of pressures we would need for us to regain our tail, but whatever it involves it must prioritize reproduction with families who tend to give birth to children with vestigial tails that can then through sexual selection be nurtured back into a functioning spinal structure. All this is a relatively natural way that can be sped up artificially, assuming laws and such on human genetic modification is ignored.
Stage 2: Humans with tails become more aquatic, gaining webbed hands and feet, probably hairless skin(or the way of the otter/seal, insulating and aquadynamic fur, depending on the climate and various other pressures), and a more aquatic variant of their tail that will either be vertically flatter or horizontally flatter. This may be forced with melting ice caps making the sea level rise and making humans more and more likely to spend time in water, while somewhat increasing the likelihood of consuming fish due to there essentially being more ocean as the water levels fill up land spaces that used to be ocean once more. An unforced development is unlikely, but can still happen with some group of tail-having humans preferring to spend a lot of time in the water and their survival depended on catching fish, aka swimming, if fishing rods and the like somehow were forgotten about.
Stage 3: The more aquatic humans, which you'd be hard-pressed to call humans at this point, merfolk would be more apt, spend much more time underwater and in fact have become somewhat dependent on the water for their survival. Their tails now have fin-like structures, as well as their hands and feet having become elongated and a little fin-like as well with more developed webbing but due to the merfolk probably still making use of tools the hands haven't become entirely fin-like. It is here where your specific mermaids come into being. There would occasionally be a merfolk who gives birth to a baby with sirenomelia, causing fused legs, and most importantly, a fused tail whose spine would do down to the ankles or further most likely. The fins of the legs would most probably be gone, lost in the fusion, with the occasional siren(what they'd probably be called) having vestigial fins coming from the mass of their lower/back body like those tiny fins of a shark and might still help them to swim in some cases. Due to the tail being most likely longer than the legs at this point in their evolution, sirens would still have a relatively functional tail beyond where their footfins would be fused, who would not be too badly afffected by their sirenomelia, and would as such create the mermaids that you're looking for, perhaps with some inheritance factor to the condition, causing sirens to generally make more sirens and as such create a subspecies of merfolk who for all intents and purposes are basically mermaids. This could also make for some interesting cultural, social, or societal interaction between the normal merfolk and the sirens, which if you choose to go that path I'll leave the details up to you. [1]: https://rarediseases.org/rare-diseases/sirenomelia/
Mermaids are very human
Evolution and embryonic building plan was already established. This merging of the legs is a quite recent adaptation of an existing body shape, to the water. It's safe to assume that mermaids (having a lower body physiology as described by Ichtus King) are actually bipedal mammals (humans, apes) returning to the sea, not a completely separate family of mammals. As Adrian pointed out in his comment, there is no evolutionary advantage of doubling and placing and merging bones in parallel, it would only reduce freedom of movement. It is not a new building plan, it is an adaptation.. so..
A mermaid embryo looks like a (scaled down ?) human embryo
Quadrupedal mammals that return to the sea tend to degenerate the legs and bipedals like humans will merge their legs ? If so, mermaid embryos will look very similar to human embryos, up until a very late stage. They have two legs, like we do. At some point, a fleece will form, the legs flesh grows together and merge, keeping the bones of two legs inside. This tail is a problem at birth of course, that is why it should be flexible and foldable when birth takes place.
There are aspects that simply don't connect, when you regard mermaids as species. A mermaid giving birth to a baby mermaid would have severe issues giving birth anyway, because the legs cannot be spread to give room. This would suggest there is no way for mermaids to give birth, without a drastic change in embryo shape. In nature, such things don't happen. It could be scaled down..
The issue of reverse-engineering mythological creatures
Maybe when biology would have been more advanced at the time, this bizarre mix between fish and mammal would never have been invented. Assyrians and Greeks did not have this forum to debunk their creations. As modern humans, we can look with astonishment at these fantastic creatures, but in fact, it may be inappropriate to pose scientific questions, or demand scientific reasoning, when talking mermaid, pegasus, sphynxes, chimearas.. at the time these creatures were invented, the combinations served a symbolic/psychologic meaning. There is no sound, biologic explanation needed, because they simply not exist.
Just not with actual vertebrae attached to the distal femur.
Your merfolk will have a distally fused monofemur as described. I'd suggest rotating the pelvic sockets posteriorly just a bit to allow a) for easier mating & easier defecation and b) for stronger anterior-posterior limb motion, which will help them swim better. They'll basically swim like a whale, but not with their tails.
Rather than messing around with the human tail, we're only going to look at the feet. Your merfolk will still have a knee, but it will lack a patella, either by deletion or rearrangement, and will be able to articulate both anteriorly and posteriorly. The calf will still have a tibia & fibula but will be extremely short and will be able to rotate along the limb's central axis. The major development of the merfolk will be in their foot. Rather than the 90 deg angle of the present human foot, they'll reevolve a planar foot either through deletion or reduction and rearrangement of the calcaneous.
The number of toes, metatarsals and phalanges will be the crowning glory of the merfolk foot. With ten digits, perhaps some extra metatarsals and as many as fifteen or twenty phalanges for the medial halluces and perhaps 10 to 12 for the lateral toes, all fully webbed, your merfolk will have a tail that not only undulates in the vertical plane but can also rotate to effect quick turning, diving and surfacing manoeuvres in any body orientation.
The patella, short tibia/fibula and extra foot bones will look somewhat like "vertebrae" attached to the femur, but will make evolutionary sense
