Many people denote by R the field of reals, not just its underlying set. IMO this is a very sensible thing to do, for the following reasons:

(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.