How "reincarnation" might work?

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 “Individual's subjective awareness”

Awareness is not a good enough criteria. For one thing awareness is combination of unbroken continuity and a little data. The data *is* important. Data can be recreated. 
But the unbroken continuity is not enough to define a person.

For example: You begin to change for some reason molecule by molecule into another person, say Mussolini, all the while maintaining your continuity. All the while your continuity is persevered and yet in the end… you are gone and there is only Mussolini. Continuity is simply not enough.

 For another thing: sleep
 
You don’t see out of your future self’s eyes (at present).  You don’t think your past self-thoughts and yet they are considered the same person

“Internally, each being would be aware that they are themselves and not the other”

Actually with the same memories they would have no idea who had thought what.

 
Having separate consciousness’s doesn't *do* anything. 
The point of Science is that practical concerns are all. What we can see. What we can feel.
We define almost everything else (machines, books, resources) by what it does. If one is replaced by a perfect copy, broken continuity is an invisible force that doesn’t affect anything. It does nothing, so it is nothing. If it does something it only does so because it affects one’s form.
 If you don’t care about it, it doesn’t matter. Consider Captain Kirk “killing” and "recreating"himself each time he beamed down to a planet.

For the record if I was replaced by a perfect copy I would not care. Heck I’d hope to get a trade up to a better looking version.

One’s Form is important because it comprises all of the things that are the things that make us what we are.
One important thing to keep in mind is that people are not set in stone. A clone would change to be different to the original but then so do you. Almost everything does change… a lot, every second from day to day.
 To be you a body just has to be as much you as you twenty years ago. 
There is a good deal of wriggle room.
They just have to have enough of your important aspects to meet that standard.
Continuity is quite a minor aspect. I mean it’s important in paying your bills and keeping your heart beating but in defining a self it is just a line through time and space like everyone else’s.

Having a clone is somewhat like having a future self time-travel. You can see you from the outside and occupy a different space. But it is still you. Simply a you, living a different life.

Why do people care so much about the body/finite consciousness?  Instinct. Millions of years of evolution have wired us to care a great deal about this body but that does not mean it is important in defining us. Wanting something is an important part of you but that does not mean that you need “something” to be yourself.
 We also have a deep fear of the “other”. Of the doppelganger and life stealer.