Much of what is discussed by philosophers of personal identity focuses on cases where it isn't clear (at least not on the face of it) whether someone is still the same person under some hypothetical change of circumstances. So for instance, if Fred wakes up one morning and has lost all his memories of the past, we might ask whether he's still the same person or whether his personality has been somewhat reduced by such a loss of recall. We might then go further and ask what we would think if it turns out that instead of his own memories, Fred now has the memories of someone else — is he still, in this changed scenario, the same person as yesterday, or should we say that he's now someone else, transferred into Fred's body? At which point would we have to conclude that we're no longer talking about one and the same person? Does it depend on how we lay out the scenario, that is, do our intuitions here depend on the kind and order of changes we gradually introduced? There is a stream of literature on these and related questions about personal identity. But it's mostly, as I said, about constructing borderline cases, scenarios in which we can test our intuitions and assumptions.
In our discussion here it's the other way round. It's not a scenario in which we are uncertain about the identity of a character at all. When I wrote that Arnold is the same person in act I as he is in act II (when he is thrown into a fictional world), that claim is a simple assumption, or stipulation. He's the same person because we take him to be the same person. The playwright supports that assumption by arranging the first switch to a fictional world so that it almost suggests itself. At the end of act I, Arnold stands in the middle of the room, when there are suddenly a thunderstroke and a few seconds of darkness — after that, the scene has changed (the telephone, as the primary passage marker, has disappeared), the other actors have suddenly appeared in different costumes. Arnold is standing at exactly the same place, in the same clothes, and gives a surprise interjection. Thus it's a very natural assumption that, while much around him has changed, he's still the same. But when we make that assumption, it's exactly that: an assumption. Arnold is still Arnold because we take him to be still Arnold.
Contrast this with the cases that are interesting to the philosophy of personal identity. There, we have some reason to ask ourselves whether someone's identity has changed, and from that we can start a philosophical reflection on the notion of (and our intuitions about) that sort of identity. In our case, however, most of the interpretation of the rest of the play is based on taking the character of Arnold as continuous. So our imagination is clearly directed to taking him as the same person throughout. It is stipulated that Arnold is still Arnold (even as he has been transferred into the world of a fiction).
So personal identity, instead of being questioned and explored in a borderline setting, is taken for granted here, and there is some effort to construct the scenario precisely in a way so that there is no question about it. In the case of Improbable Fiction, this is competently done, and it simply works. Obviously, there might be a limit to such a construction. If, let's say, the play wasn't a play, but a novel, and instead of simply being transferred to somewhere else, Arnold would also find himself in a completely different body, perhaps even the body of an animal, say, a crocodile, plus fully unable to remember anything from earlier times, and incapable of speaking and thinking at all ... well, you get the picture — at some point the stipulation that this is still the same character, Arnold, would become unintelligible. And now we might ask ourselves where exactly that point is. Is it when Arnold has become an animal? (But then, Kafka's Metamorphosis seems intelligible.) Is it when he cannot remember anything anymore? (But would we say that someone ceases to be a person just because they suffer amnesia?) And so on. We might explore our intuitions about such cases. We might do some 'experimental philosophy' to find out empirically how widely exactly which intuitions are shared. And so we might learn something interesting about our concept of personal identity.
But none of this really plays a role in our understanding the plot of Improbable Fiction (and similar plays, movies, or novels), and the continuity of a character in it. And that's because that continuity is essential to the device of passage into an instance of unreality. So an author has to simply stipulate such continuity, to make us naturally assume it. He'd only ask for trouble if he would stage it as problematic. These plots are not designed to discuss the question of personal identity (as the scenarios in the philosophical literature are). Rather, they're designed to avoid or preempt that question. Thus it doesn't seem to me that we can learn anything interesting about the philosophy of personal identity from them.
 The notion of stipulating identity, rather than discovering it, is the same here as that on which Saul Kripke insists when he criticizes the idea, vented by some, of 'trans-world identification' in the first lecture of his Naming and Necessity. (I'm as much an antirealist about possible worlds and fictional worlds as Kripke, just in case there was any doubt.)