Big Lipped Alligator Moment: While Snake is hiding in the sauna, waiting for the brooch to change shape, a naked guard enters the sauna, hangs out a little bit, and then leaves. Granted, the guard is probably there to ensure the player keeps Snake in the sauna the right amount of time, but it's still a pretty random moment.
Even Better Sequel: Or Surprisingly Improved Sequel for those who weren't impressed by Metal Gear. This game features a more complex plot, better graphics, the introduction of the radar, the ability to crawl, enhanced guard AI, and tons of little details. The improvement is so vast that first time players have been stunned by the opening credits sequence.
Growing the Beard: This is where the Metal Gear franchise starts to resemble the Metal Gear franchise.
Harsher in Hindsight: Calling Miller during the fight with Big Boss will have him state that Big Boss was a monster. Granted, it's in the context of Big Boss's Super Strength, but this line becomes significantly harsher after Peace Walker, where he was a former ally of Big Boss and one of the founders of Outer Heaven.
Likewise, calling Kasler during the fight has him mentioning a false rumor that Big Boss has cybernetic limbs to replace the ones he lost during the fight in the previous game. Metal Gear Solid V: The Phantom Pain revealed that he did lose an arm and had it replaced with a mechanical one, long before the events of the first game. Of course, that wasn't actually Big Boss. But in an almost hilarious turn, the "Big Boss" of the first game, Venom, the Body Double, did have a bionic arm.
Big Boss's comment about soldiers being dead weight out of the battlefield becomes somewhat hilarious after watching the Mega64 trailer on Metal Gear Rising: Revengeance, where Raiden bombed at absolutely every single job his unemployment adviser offered him up until getting a job as a cybernetic private military contractor. It also goes int a bit of Harsher in Hindsight, as the unemployment manager when first meeting Raiden mentions attributes to Raiden's 4 year unemployment to being the result of an implied terrible economy (at the time the trailer was released, the American economy, not to mention most of the world wide economy, especially the European economy, was undergoing a severe economic crisis).
Memetic Mutation: The phrase "It's not over yet!", used in every Metal Gear Solid game released after this, excluding the Acid games and Ghost Babel.
Gustava's backstory reveals that she fell in love with a man named Frank Hunter, whom she intended to elope with and cross the border with him. Unfortunately, she was denied permission, and was outcasted as a result. Her death by Metal Gear D hits Snake hard when he reveals that he didn't want to see a woman die in front of him in Metal Gear Solid.
For Gray Fox, he's a Shell-Shocked Veteran who cannot stay away from war as he thought he cannot function outside the battlefield. As the pilot of Metal Gear D, he ended up killing the woman he loved. He thought making a woman happy is something he cannot do. The theme that plays in the conversation between him and Snake is appropriately called "Tears".