Here are some additional arguments I have encountered, in various sources, against zygotes and embryos being living individual organisms, which I will review and judge on their own merits here, as well.
One of the most popular arguments, widely believed by many, including some in the scientific community, is that, prior to fourteen days after fertilization of the oocyte by the spermatozoon, the embryo is not yet an individual because there is the potential for monozygotic twinning to occur, causing there to be two individuals instead of one. This argument assumes that this split into two individuals erases the existence of the original embryo, leaving two progeny in its wake. However, in reality, it is thought that monozygotic twinning occurs at the blastocyst stage of embryonic development, in which the cells of the inner cell mass have separated from the cells on the outside of the embryo, which form a structure called the trophoblast. When monozygotic twinning occurs, part of the blastocyst separates from the rest of the embryo, splitting off and giving rise to a genetically identical clone, or twin. It is very important to note here that this process does not erase the existence of the original embryo, and, in fact, due to the embryo's amazing ability to heal its wounds and regenerate missing cells, it actually makes a pretty decent recovery afterwards. Neurobiologist Maureen L. Condic compared this process to the analogy of an adult human's arm being cut off and used to create a clone of itself, while the original is able to regenerate its missing arm afterwards.
Indeed, the very mention of cloning allows me to segue into the mention of the fact that, as human cloning, by the merging of any reprogrammed somatic cell with an oocyte, is at least hypothethetically possible, the truth is that you or I have the potential to be cloned, which is basically the same process as monozygotic twinning, at any moment. Therefore, if we utilize the argument against individuality from twinning/cloning, then, no adult animals, including humans, are ever individuals, as they could, potentially, be cloned at any time. This is obviously an absurdity, which means the above argument must be, as well.
Another argument, this one based more on lack of information than anything else, really, is that, as embryos are capable of being frozen and thawed back out many years later, emerging alive, while this has not been done to adult animals yet, this proves that embryos are less alive than adult animals. Yet a simple exploration of what actually happens during the embryo freezing process dispels this one entirely. During this process, water is expelled from the cells, as, since water forms sharp crystals that penetrate and kill cells when it freezes, it is dangerous and deadly to allow an organism to freeze without first doing so. Then, antifreeze is put into the cells in place of the expelled water. The only reason why this has, to date, been done successfully only on embryos, and not on adults, is simply because it is far less practically feasible to carry out this process on an adult organism, simply because the latter is so much larger than an embryo. It is only a matter of practicality based around physical size. There is nothing inherently different between an embryo and an adult that causes this difference. Who knows? Perhaps, in the future, preserved, frozen adult humans will be a reality, just like preserved, frozen embryonic humans are now.
Lastly, there is the argument that, since the trophoblast forms what are commonly referred to as extraembryonic tissues, including the placenta and yolk sac, while the inner cell mass forms what is thought of as the embryo proper, before the separation of the trophoblast from the inner cell mass at the blastocyst stage, the embryo cannot yet be an individual. However, a closer examination of this argument reveals critical faults. The fact that the structures formed by the trophoblast are referred to as extraembryonic structures is rather misleading; in reality, they are, indeed, part of the embryo's body, just like what is thought of as the embryo proper. The fact that they are utilized only during the antenatal stage of life, and subsequently shed upon parturition, does not make them any less part of the embryo's body than the fact that milk teeth are utilized only during childhood, and are subsequently shed, makes them any less part of postnatal children's bodies.
Overall, these three additional arguments against zygotes and early embryos being individual organisms can all be soundly rejected. The argument from twinning can be rejected due to the fact that twinning only produces a new embryo, while the original remains, and that any adult organism could potentially be cloned at any time, the argument from freezing and preservation can be rejected due to the fact that the difference between adults and embryos in this respect is only a matter of size, and nothing more fundamental than that, and the argument from extraembryonic structures can be rejected due to the fact that these are, indeed, parts of the embryo's body, which are subsequently shed after birth.
Condic, Maureen L. (2014). Totipotency: What It Is And What It Is Not. (https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3991987/)