You're right, it's not really the product of some set "G" with itself but hooooo boy a programming/hacker forum is probably the wrong place to be defending the syntax of mathematical notation
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..?