Feels like Brosnan Era Bond
The 007 they depict feels thematically and stylistically most similar to Pierce Brosnan's Bond. The action and vehicle sequences remind me the most of the 1990s films, but Gibson's Bond has a grace and elegance to him that's very reminiscent of Brosnan — the general tone and personality feels very much in line with Brosnan's Bond as do the locations and premise of the game.
The combat looks a lot like Daniel Craig's Bond, who is very much a street brawler. I also see some resemblance to Tim Dalton.
There doesn't seem to be a lot in common with the Connery or the Roger Moore eras, except maybe the gadgets and Bawma kind of feels like a Moore villain, but I'd say more like a Craig one.