Kripke's argument for a casual-historical view of names is first and foremost about proper names. It is a theory to explain how proper names are used in natural languages. Kripke argues that rigid designators are central to our use of language and how we, in this world as opposed to a possible world, use language. To illustrate, from Marc Cohen:
The claim that a designator is rigid does not mean that we can’t imagine a possible world in which that designator is used differently from the way it’s actually used. We can imagine a world in which a different set of parents named a different person ‘Benjamin Franklin’. But that would not be a world in which someone else was (as we use the name) Benjamin Franklin.
As Kripke stresses, it’s the way we use the designator in question that determines which object, in a possible world, it designates. The way we use the expression ‘the inventor of bifocals’ (attributively) it designates, with respect to a given possible world, whoever invented bifocals in that world. So in the actual world, it designates Franklin. In another possible world, it designates Spinoza. In yet another possible world, it designates no one. (Imagine a possible world in which there were no bifocals, or in which they were found growing on trees.)
There is a possible world where some other person is born to different parents and they name him "Napoleon" and he then goes on to do the things that Napoleon did in this world, but that would still be a different person altogether. This is because it is not the same person that we refer to when we use the rigid designator "Napoleon." The initial baptism is truly like a literal baptism, it is the specific moment in time when the baby is born and the parents first refer to it by whatever name. After that point, in all possible worlds, any event that takes place in that child's life is attributed to that child by way of a casual-historical reference chain. If a different baby was born to different parents it would be a different initial baptism and therefore our language would not be referring to the same person.
Another important point to consider is that Kripke is the ideal of essential properties. From Lecture I of Naming and Necessity:
Let's call something a rigid designator if in every possible world it designates the same object, a nonrigid or accidental designator if that is not the case. Of course we don't require that the objects exist ill all possible worlds. Certainly Nixon might not have existed if his parents had not gotten married, in the normal course of things. When we think of a property as essential to an object we usually mean that it is true of that object in any case where it would have existed. A rigid designator of a necessary existent can be called strongly rigid.
So clearly Kripke does argue for a sense of essential properties and one of those properties is whom the parents of the person are. In Lecture III of N&N he does address the question you are asking in regards to the Queen of England:
Let's try and refine the question a little bit. The question really should be, let's say, could the Queen-could this woman herself-have been born of different parents from the parents from whom she actually came? Could she, let's say, have been the daughter instead of Mr. and Mrs. Truman? There would be no contradiction, of course, in an announcement that (I hope the ages do not make this impossible), fantastic as it may sound, she was indeed the daughter of Mr. and Mrs. Truman. I suppose there might even be no contradiction in the discovery that-it seems very suspicious anyway that on either hypothesis she has a sister called Margaret-that these two Margarets were one and the same person flying back and forth in a clever way. At any rate we can imagine discovering all of these things.
But let us suppose that such a discovery is not in fact the case. Let's suppose that the Queen really did come from these parents. Not to go into too many complications here about what a parent is, let's suppose that the parents are the people whose body tissues are sources of the biological sperm and egg. So you get rid of such recherché possibilities as transplants of the sperm from the father, or the egg from the mother, into other bodies, so that in one sense other people might have been her parents. If that happened, in another sense her parents were still the original king and queen. But other than that, can we imagine a situation in which it would have happened that this very woman came out of Mr. and Mrs. Truman? They might have had a child resembling her in many properties. Perhaps in some possible world Mr. and Mrs. Truman even
had a child who actually became the Queen of England and was even passed off as the child of other parents. This still would not be a situation in which this very woman whom we call 'Elizabeth II' was the child of Mr. and Mrs. Truman, or so it seems to me. It would be a situation in which there was some other woman who had many of the properties that are in fact true of Elizabeth.
When he says "There would be no contradiction, of course, in an announcement that (I hope the ages do not make this impossible), fantastic as it may sound, she was indeed the daughter of Mr. and Mrs. Truman." what he means is that there would be no logical contradiction if we find out that it is in fact true that the Queen of England was born to different parents than the people we think she was born to now; it is a possible proposition. What he is saying is that "Elizabeth, Queen of England" in our natural language refers to that woman there, sitting on the throne. If it were the case that we lived in another possible world and there was some other woman who was born to different parents whom also became the Queen of England, even if she had the same name, Elizabeth Alexandra Mary, we would be referring to a completely different person when we referred to her.