Apologies in advance for the length of my response. A simple answer will not suffice.
The phenomenon described is chimerism. Most simply defined, chimeras are animals or humans whose cells are derived from two or more zygotes. Their bodies contain two different sets of DNA. This is important! We will come back to it!
Chimeras are derived from two or more zygotes. The vast majority of human chimeras are single persons, not twins. Their chimerism is usually something like having two different blood types or other genetically different tissues.
In fact, human chimeras are exceedingly rare and most go undiscovered. The abnormality would generally only be found by genetic blood or tissue testing. Also rare, but more dramatic, are sex chromosome chimeras which give rise to an individual who is a true hermaphrodite or has ambiguous genitalia.
In answer to the specific question: When a zygote splits into two zygotes that would then go on to form identical twins, those two zygotes are identical in all respects including their DNA (chromosomes). They are two identical individuals and, according to current Church teaching, each would be ensouled.
In the scenario where a zygote splits into two and then fuses together, one can reasonably assume that, when in the split stage, each could have some differences in their DNA. It’s a reasonable assumption in that we know the subsequent chimera will have at least two different sets of DNA in the now fused single zygote. By definition the DNA would be different as that is the definition of a chimera. These differences in the DNA would be most commonly due to fusion with a fertilized polar body. (A polar body is the byproduct of an oocyte meiotic division.)
Because of this, it may not be unreasonable to speculate that there is an abnormality which led to the fusion. In the two-cell state there may have been a “normal cell” and an “abnormal “cell”. The normal one being complete in its genetic complement would fit the criteria of personhood and be ensouled. The abnormal cell would not.
In an article as recent as 2020, true twin human chimeras have not been found. So, for the moment, the question as posed by the reader may be moot. However, if we are to be open to the idea that true human twin chimeras may exist, then the current Church teaching that an ensouled individual person exists from the moment of conception will certainly generate renewed debate on the subject.
It is worthwhile mentioning that in 2007, Pope Benedict XVI asked for an update to the 1987 Instruction Donum Vitae. It was apparently his view that within that 20-year time span, new medical techniques had raised ethical issues that were to re-addressed by the Magisterium.