Why don't we go with genre recommendations? Two of each, limiting myself to an initial release date mostly on or before 2000!
FPS
Blood - One Unit Whole/Fresh Supply
If I were to name a few FPS games that defined the retro era, Doom would be at the top of my list. But this is a very close contender. A balls-hard shooter that is a love-letter to contemporary horror movies, with unconventional weaponry, a cast of colourful horrors to blow up and believable labyrinthine level design, this is one of the few shooters that aged very well.
Doom (Every single iD Tech 1 Doom game)
Yes, lumping Ultimate DooM, DooM II: Hell on Earth, DooM 64, Final DOOM, and every
Official Add-On here, because the zeitgeist of DooM is not limited to or segregated to any single game, but an amalgamation, a terroir of everything classic Doom that had ever existed.
What I really love about classic DooM is that it sets the trend of the standard loadout of weapons in an FPS game, a trend set for a good reason. With every weapon serving a specific purpose, doing well for specific situations, and a roster of enemies that deliver a good challenge, this is a gameplay style that will never get old. Moreover, with the massive library of add-ons, mods and source ports available, you'll be enjoying Doom classic for a very long time.
Action Adventure
Burning Rangers
This one calls back to fond childhood memories of mine. Back when we were given the opportunity to pick a new gaming console, we went with team Sonic, and we did not regret it. While the PS1 carried with it far more gems in the RPG department, the Saturn excels in arcade ports and fighting games.
And it is games like this that captured our, specifically Sophia's, imagination at an early age.
Legend of Zelda - Link's Awakening DX
Picking the N64 Zelda games would probably be more relevant, as a lot of folks have had fond memories of this game, but it would not be honest. For us, our best experiences with classic Zelda was sitting in front of my PC, with a GBC emulator, and doing everything there is to do in Link's Awakening DX. It is for this reason that I would recommend LoZ:LADX.
Platformer
Sonic 3 & Knuckles
Given that I picked team Sonic back then, I ended up spending my early teenage days firing up Sonic Jam, and playing through every included Sonic game repeatedly. Over time, we realized how poorly Sonic 1 aged, and while Sonic 2 was a great improvement, the lack of content means that eventually, I'd be able to run through the entire game in a couple of hours at most, and get bored of it, even with the option of playing as Knuckles in Sonic 2. This left Sonic 3 and Knuckles as the clear winner out of the classic Sonic games for me.
Castlevania: Symphony of the Night
When I started to get into PS1 emulation, I ended up listing a lot of games that I had wanted to catch up on, and this was one of them. I did not regret playing through this game, as the non-linear/exploration aspect of it really did a lot to elevate this game above its predecessors in a way that only Aria of Sorrow for the GBA could top.
Strategy
Super Robot Taisen Alpha Gaiden
One of the gems that came out of the ROMhacking and fan translation community. This Strategy RPG is an unabashed fan service to all mecha anime fans, and leans into the premise of the story being a massive crossover, while creating a story that is compelling in its own way.
Final Fantasy Tactics - War of the Lions
With a name like FFT, you wouldn't think that this game would have such mature writing to it. A solid entry to the world of Strategy RPGs, this game really aged well, especially with the choice of its art style.
RPG
Valkyrie Profile: Lenneth
My first experience of this game was when I was visiting a friend for a weekend of gaming. We were teens, and while he spent the day jacking off on the couch to female tennis players on TV, my ace ass was captivated by the...everything that was Valkyrie Profile. The music, the art style, the gameplay, and the story premise was nothing like I have ever seen before. This might have been the moment when Motoi Sakuraba became my life long inspiration as a composer.
Grandia II
Our first experience with this game was during a particularly dark era of our late teens. This game is surprisingly mature, touching on issues like childhood trauma, religious trauma, personal tragedies, and plurality, making it a seminal piece of work that not just captured our imagination, but also informed a lot of personal philosophies and spiritualities that stuck by us to this day. The gameplay is also really good, and features one of the best implementations of the ATB system I have seen to this day.
Racing
Wipeout 2097
Let's be honest here. I am not a fan of racers, at least the more realistic ones, I feel like it is a genre of game that I'd enjoy watching more than playing myself. However, when I first played Wipeout 2097, it blew me away, in more ways than one. Not only do you have the acing aspect, but you also have the combat aspect, with parts of the vehicles that can get damaged or even totaled by collisions and weapons, adding a more dynamic, tense and exciting to this game, especially with the different kind of physics. All of this, while trance and breakcore music pounds in the background, made this racing game a cut above the rest.
Re-Volt
In a similar vein, this is an RC racing sim with complex physics that gives it a unique feeling. But what sets this game apart from most other racing games is that it also has combat aspects, with a set of powerups that not only give a wide variety of effects, but with a couple of them even leaning into the psychological warfare aspect. (Bomb and Decoy powerups) With these traits, it was a toss-up between this game and Star Wars Episode 1: Podacer, but with Re-Volt, giving you the ability to create and race on custom tracks, I am nominating Re-Volt here.
Fighting
Street Fighter Alpha 2
This game was the reason I chose the Sega Saturn over the PS1, the 6 face button configuration o the controller made controlling the characters in this game quite a dream. I have spent so many sleepless nights repeatedly playing through the arcade mode with various characters and experimenting with various combos. It was probably a big influence on, funnily enough, leading us to obsess over the Tales of series, when we started to get into RPGs. I still think that this game is worth revisiting if you're looking for a retro fighting game.
Fighters Megamix
A cross over of the roster from Fighting Vipers and Virtua Fighter, this game features, at the time, a fairly unique and new way to control fighting game characters in a 3D environment, which makes it a fun and evergreen experience to just set up custom battles with various characters and arenas. The music, and clothes/arena damage mechanic is also pretty cool.