This is an interesting medical ethics question and has been dealt with in many books on the subject, but it remains controversial. The best example of this ethical dilemma is the case of conjoined twins, but where one twin has not developed beyond the size of a small lump on the forehead of the other twin. The two are genetically separate, but the tiny twin depends completely on the larger twin for survival, because it has no heart, and no developed brain, but on cell sampling is shown, for example, to be of a different sex on genetic analysis. Is it morally acceptable to remove the lump? If the lump was an 'identical' twin, we would not know, because unless it had its own separate anatomy, it would be otherwise indistinguishable by any genetic test we have. Is it immoral then to remove the lump?
One solution is to treat the two as we do a mother and fetus, which can be defined as two genetically distinct beings, and therefore the lump cannot be touched, but deserves Baptism and its own name. This is more or less the Catholic approach. The counter argument to this is that it is possible for a mother to have a fetus which is her genetic clone (albeit not naturally, but identical twins are precisely such clones). Are they two different people with two different souls? The obvious answer is yes, but it raises the question of what is human life, and what is not.
Cancers, for example, are autonomous forms of life that have developed a genetic dissimilarity to the host at some stage beyond that of the zygote and take on a parasitic existence. Is the cancer part of the person? Is it human? Probably not, but again one can ask, what does it take to be human? Can you be a twenty year old lump on someone else's forehead, and be human? I think in most cases we never know it's human and it gets removed inadvertently. This is morally neutral because a sin requires knowledge and consent.
So, my answer is that, for practical purposes, a twin has its own soul once it is clear to the observer that it is a distinct genetic or otherwise autonomous entity that is nonetheless human (conjoined though it may be). We have an obligation to treat it as a person once this becomes obvious. There is also an obligation to answer to the question definitively once a suspicion is raised that some appendage to a person may in fact be the person's twin.