In addition to the twinning argument, one additional argument sometimes utilized to deny the individuality of early embryos is the fact that two embryos are capable of fusing together to form a single organism. This process is known as tetragametic chimerism, and the resulting individual is referred to as a tetragametic chimera, or simply a chimera. They are called tetragametic because they originated from four gametes, twice the number as someone who is not a chimera.
The argument asserts that, as two embryos have the potential to become one individual, this means that, before fusion, each embryo cannot be regarded as a single individual in its own right. However, I find this argument to be as jejune and flawed as the twinning objection, and I will elucidate why I think so.
Just like how I mentioned that the twinning argument is rendered absurd by the fact that any adult animal could potentially be cloned, which is basically delayed monozygotic twinning, and, in fact, has even been referred to as such in the peer-reviewed scientific literature, as shown in the example cited below, I think that the chimerism argument is rendered absurd by the fact that organ transplants between adult animals are, in fact, not just theoretically possible, but already happen quite routinely.
As a hypothetical gedankenexperiment, let us envision a scenario wherein half of one adult human's organs are defective, and urgently need to be replaced. Now let us say that half of the organs from another adult human's body are removed, killing the unfortunate donor in the process, and transplanted into the recipient, with the result that the recipient now has half of the organs in their body originating from someone else, and comprised of cells with a different genome, rendering them a postnatally-derived tetragametic chimera.
In this scenario, no one would deny that, prior to the fusion, there existed two distinct individual adult organisms. Likewise, the same would hold when this process occurs involving a pair of early embryos coalescing into a singleton.
While, for ethical reasons, such a scenario is obviously unlikely to happen, it still means that, at least in principle, it is possible to form tetragametic chimeras in adulthood via the process of organ transplantation, just as, at least in principle, it is possible to form monozygotic twins in adulthood via the process of cloning.
Therefore, just as the fact that cloning is hypothetically possible at any age of postnatal life renders the argument that the ability of a single embryo to split into twins during the process of monozygotic twinning means it is not yet an individual absurd, so, too, does the fact that extensive organ transplantation is hypothetically possible at any age of postnatal life render the argument that the ability of more than one embryo to combine into one during the process of tetragametic chimerism means that neither are yet individuals absurd.
Med Wieku Rozwoj. "Human clone or a delayed twin?" 2001;5(1 Suppl 1):39-43. (https://www.ncbi.nlm.nih.gov/m/pubmed/11684762/)