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The G-band was never part of the standard Johnson-Cousins system, and so my understanding is that it’s not particularly standardised. In fact, professional observatories now tend to all have their own bespoke sets of filters, and so you can’t directly compare a PanSTARRS g with an SDSS g, for example. Though they’re very similar.
As I understand it, the Gaia G-band is indeed the unfiltered response of its CCD. Though, of course, Gaia doesn’t have any atmospheric extinction, so it won’t be a perfect match to anything you can get on the ground.
If one is being really pedantic, there are multiple different revisions of the Gaia G-band, as each data release includes improved instrumental corrections (which depend on colour). If you google for Gaia photometry you can find plenty of long gory papers about the process.