(G) stands for from Pokémon GO.
To be specific: Origin Game = GO, Location = Pokémon GO.
As a result, only direct GO->HOME transfers have them, not GO->LGPE->HOME
(GO->LGPE->HOME location ends up being Go Park Complex. As I briefly touched on above, location data also matters.)
I'm addressing why GO mons have the G-mark. (Why it's important to have a visual distinction for GO mons)
I don't think this was officially touched on, but I believe it's in part due to the HOME->SWSH travel restriction placed on GO mons.
At a glance, you need to know what mon may not be able to enter SWSH, and why.
While they do place a on the mon, to show it can't travel into SWSH,
If a Mew has it, one may be confused: isn't Mew in SWSH, why can't it go into SWSH?
GO travel restriction: for legendaries and mythicals, if you haven't had it in SWSH, you can't transfer that species into HOME (originating from GO).
[Present day workaround, as long as the legendary/mythical doesn't bear your HOME's TID/SID combo, it can enter SWSH just fine. don't know if it's a bug that TPCi/GF will patch]
(in my tweet, I did call it row, when I meant column)