It’s quite possible you’ve seen photographs of mermaids. They are not real. They are forgeries where a person took dried out limbs of a monkey and sewed on parts of a fish. The first one to do this, (P.T. Barnum who leased it from someone and) took it as a traveling circus exhibit which earned a nice income for many years.
But in truth the belief in mermaids goes back to early history. Sailors claimed to have seen mermaids and it can be assumed that walruses or other large sea mammals were seen from afar or underwater and thought to be mermaids. People fed this idea with the finding of strange animals. For example the Rhinoceros was described by people as having metal armor. Even the wisest people relied on information from tourists and sailors for that was the sole source of information available.
Now back to the mermaid, does it exist in the words of our sages?
Our Talmud does not talk expressly about mermaids. The sole source people point to is where the sages describe a “dolphin”. In Tractate Bechorot 8a it says; “The dolphins procreate like man. What are dolphins? Rav says the children of the sea.”
Many would like to interpret this Talmudic passage as referring to mermaids. The main commentators who say this are Rashi, Tosafot and the Chida. But they never described creatures that could talk and possess a soul; rather they described sea creatures that have their outer appearance look like half man and half fish. Other commentaries explained according to the simple understanding that dolphins (the ones we know) are mammals that nurse their young and this is the meaning of “procreate like man”. (See Mosaf HaAruch under “Dolphin” and Rabbi Yaacov Emden and the Yalkut Shimoni)
In another place our sages talk of a creature that looks human called “Adnei hasadeh” “master of the field”. (Tractate Kilayim, Chapter 8, Mishna 5) There is no doubt that there are very strange creatures in our world some even have a semblance to man like the proboscis monkey and the orangutan.
It’s quite possible that just as there are land animals that look similar to man there may also be sea animals that look similar to man. It should be mentioned that even those commentators who talked of “daughters of the sea” did not talk of intelligent creatures that talk and think like man rather just another species of animal. Some say they could sing, but so can birds. So even according to Rashi and the Chida there is no basis to say these were talking humans that live in the sea, rather they are just sea creatures that look like man.
In the Tractate Chullin 127a, our sages wrote that corresponding to all the land animals there are also a parallel set of animals in the sea. “Whatever is on land is also found in the sea.” So it is quite possible that we will one day find a sea creature that looks like a man even if only half a man. Even now there are fish species that have faces that look human as we saw.
Are there creatures whose upper half looks like the human torso? Meanwhile there is much more we don’t know about the sea creatures that what we do know. Even giant creatures like the giant squid, which has some scientific evidence for its existence and is estimated to be over 14 meters long, still hasn’t yet been sighted.
At any rate, even if sea creatures that look like man are eventually found this won’t change science. After all there are many types of monkeys that bear resemblance to man and many other creatures that look like each other. As far as your question about mermaids having a soul our Torah says that there is only one creation that has a G-dly soul and that is man. As it says and He(G-d) blew into his(Adam’s) nose a living spirit”. (Genesis chapter 2, verse 7). So even if there are creatures that look like man they are only creatures and do not have a soul like man.