(0) When you construct the reals, you don't construct just a set - you construct a more elaborate structure into which the rationals, equipped with their ordered field structure, can be embedded in a structure-preserving way.
(1) When you work with the reals, you again don't just use the underlying set. If you only need the underlying set, why couldn't you just take the power set of the naturals? It's much simpler to construct, and qua set (that is, taking only cardinalities into consideration), it's just as good.
And, of course, every field can be regarded as a vector space over itself in an obvious way. So the notation R^2 is completely justified.
OTOH, I completely agree with your complaint about the notation G^2.
I agree that you've found two notational abuses but I think they're very common; as an example of the second, the sphere S² is not the same as the torus S¹xS¹.
Programming languages are praised by their elegance and economy, even when they're wacky. Any theory (programmatic, mathematical, scientific) benefits from those qualities, because theories like to be axiomatic and terse. But any primer or intro or lesson is made MUCH better through repetition and redundancy. Computers don't need the rule of three, people do, and there's an mismatch.
The reader's walking in with knowledge of the product-based construction of N-dimensional space, and we readily admit that "R^2" is (a set literally, but) taken to mean the space equipped with whatever operations appropriate for whatever algebra.
I'm being kinda sophist on this. "G^2" isn't meaningful. I agree. But given the convention of naming a set with the implication of whatever topology/metric/norm, the notation is already under harsh torture, right..?
A superscript can refer to other things besides exponentiation, and using "R^2" to refer to the vector space is a perfectly valid thing to do.
Now I know mathematicians often refer to spaces by their underlying set, but the notation G^2 doesn't really make sense (for starters R^2 = R x R, what is G?). Calling it the Geometric Algebra over R^2 would be fine. You can even refer to it as just R^2, as long it's clear that you're using the geometric product (and what metric you're using).