After looking at all the answers - many of which are noble - I think the best answer is a synthesis of the answers proffered here with a few minor tweaks.
I think Andrei comes very close to a complete answer and there is nothing per se that I find disagreeable, but I think more can and should be said. Likewise, Suminda's answer is straightforward and seems largely correct to me and I appreciate how direct it is.
Some of the answers dwell on the word 'reborn' which exists only by convention without one iota of inherent meaning. I don't think dwelling on the particular translation of the words in the Pali having to do with 'birth' and 'rebirth' and 'reborn' is revealing in this case. Often times I think dwelling on the translations of suttas and getting attached to and wrapped in the meaning of words as if they have inherent meaning can act as a hindrance to true understanding of the Dharma. It is not that definitions of words or accuracy in translation is unimportant - indeed it often is crucial! - but getting hung up on them can also be problematic in some contexts.
So here goes an attempted synthesis...
Is the rupa skandha of Melinda and Abigail the same or different?
Different! But only conventionally so. It is stated right in the hypothetical that no two atoms are the same. Insofar as you believe in the quantum theory of matter, then you have to grant that they are different. However, this is only conventionally so because both utterly lack inherent existence.
Has it been reborn?
Sure! That's an applicable use of the word 'reborn' in this context in that I think it is useful and conveys meaning as both Andrei and Suminda similarly attest. What do we mean?
That the skhanda is in a state of constant change and flux. The body is ever changing. You can not find any two moments - of any duration whatosever - where no distinction can be made between them. In this sense, the body is 'reborn' continuously from moment-to-moment.
Of course, you could object and say that we don't normally use the word 'reborn' in this context, ie, that it is not conventionally used this way... and sure I'd grant that, but the fact that both Andrei and Suminda understood leads me to believe this convention is of course subject to change ;)
What was the manner of its rebirth? How did it occur?
It was reborn due to being impermanent and in a state of constant flux. All conditioned things are such and the manner is such and occurs because of the utter lack of inherent being in these phenomena.
How many times did it occur between 1982 to 2020?
It is impossible to find two moments - of any duration whatsoever - where no distinction can be made between the rupa skandha of one and the other.
Is the same true of the other skandhas... have they been reborn?
Are they reborn due to 'identification' with an 'I' or due to physical laws or some combination?
They are inanimate and not sentient and have no capacity for 'identification' with an 'I' no more than fire does as Andrei rightly points out. It is our identification with them and ignorance of them - as sentient beings - that mistakes these phenomena that exist only by convention and mistakes them for real and substantial phenomena. They utterly are not.
Did 'Melinda' die sometime between 1982 and 2020 and get utterly annihilated?
No! We don't say conventionally that 'Melinda' died, but rather say that she changed her name to 'Abigail'. Of course, we could change the convention to say that someone 'dies' when there name is changed. That's up to us as shared creators of our conventions.
However, to think that she was utterly annihilated is to posit that 'Melinda' existed in some real and substantial way when she utterly did not. 'Melinda' was just a label that we attached by convention and we simply changed it to 'Abigail'... This is really, really, important to understand. Thinking that anything was utterly annihilated is making the error in assuming that there was something real and substantial before! There never was!!
Was Abigail born for the first time from scratch between 1982 and 2020 and just pop into existence from nothing?
Nope! All that happened was we changed the entirely conventional label 'Melinda' to 'Abigail' and we don't call this being 'born' ... although if we all got together and started to do this ... ie, change the convention ... we could! In other words, there is nothing inherently existing about the word 'born' here that would preclude it from being used affirmatively in this situation. We just don't happen to do that as a convention. But conventions change of course. Can anyone guess why conventions can change? Because they don't have one iota of inherent existence of course! :)
It is entirely another thing to think that something real and substantial would miraculously pop into existence with the simple act of changing a name. It is entirely because there is nothing real and substantial about a name that allows us to change it. It is just a mere convention. Like all phenomena.
How is it possible that beings are reborn from moment-to-moment? In what manner and to what extent?
In the exact same way and in the exact same manner as Melinda/Abigail's rupa skandha and in the exact same way and exact same manner as all phenomena: due to the utter lack of inherent existence in any existing thing. Because all things only exist conventionally without one bit of inherent being whatsoever. If things had even a small piece of inherent being, then change would be impossible and these things would not be 'reborn.' Fortunately, this is an impossible mode of existence and hence things are reborn from moment-to-moment aka they change and are in a constant state of flux.