The events of the game is actually a VR Simulation played by Raiden.In Metal Gear Solid 2: Sons of Liberty, Raiden has done VR simulations of Outer Heaven and Zanzibar Land. No one outside of Colonel Campbell and Snake knew about the reveal that Big Boss is Snake's father. Not even Snake's support staff in the game itself (except Kaz, who knew everything since Metal Gear Solid V: The Phantom Pain). The game's lack of reveal, despite being acknowledged in the third game, may have something to do with making sure no one else knows about who Big Boss was outside of being Snake's former Commanding Officer.
The events of Metal Gear 2: Solid Snake as we know them are actually a movie based on the (in universe) real event of what happened in Zanzibar Land.In Metal Gear Solid, Campbell states that he and Snake are the only ones who know the real truth of what happened in Zanzibar Land. Couple this with the lack of Big Boss' revelation to Snake within the game and the original version of the game's use of celebrity portraits for the characters. The reason that Snake looks like Mel Gibson and Big Boss appears as Sean Connery amongst others is that they are being portrayed by those actors in universe.