Very good analogy, though I'm completely lost what is what, but it's probably because it's too early here :D
Can we all agree on the concept of 3 though? :) I mean PhD dissertations are written and successfully defended each day pro-and-contra 3.
Do we have a shared concept? Is 3 the same for a research mathematician and a 5 year old?
In practice if it works it works. The same with colors, or music, or ... well, any sort of thing and associated qualia. After all when you say 1/3 + 2/3 most people will say 1 (hopefully, though I have no idea how well known fractions are), but those pesky programmers might just laugh and say that you can't even represent them accurately.
This shows that integers (and the natural numbers) and basic arithmetical are so old in our culture that they truly are probably shared with 99.9% of humans (or let's say English speakers). But if we step a little bit outside of that by taking 4 and 8, then you might like them because they are nice powers of 2, but some Chinese people might not like 4 because it's association with death (but do like 8 because it's association with wealth & success).
And so the question is: the mind that wakes up is the same as the one that went unconscious? Is there even an abstract concept of our minds? (How tied minds are to their physical representation?) Solipsism states that despite everything indicating that there are things independently (materially or not) outside of our mind it doesn't matter because our mind is truly the only thing we have, we are, we'll ever have. (It can be a complete simulation, it can be just a dream, we can never know!)
Joscha Bach's "computationalism" (maybe he doesn't call it that) says that consciousness is evolution's answer to "how to regulate the organism in relation to other self-regulating organisms", so it's a self-directed attention of intelligence (which is also a model of things that we pay attention to). In this regard consciousness is just as part of the population as the genome, so while there are definitely you and me, but at the same time it's the same "evolutionary software" just running on a big shared pool of brains. (Which also means that we could compare every concept via computation - eg. enumerating its properties (and so on the properties required for those properties of course, so infinite regress creeps in), and this means that we can have a shared concept of 3, if it's computationally equivalent.)