Age of Empires 2.

I absolutely suck at multiplayer, but it really is a fun game. I liked how its campaign mode teaches you history too. It got two remasters, and both are quite good in their own way. 2013 has visuals similar to the originals, and Definitive has its visuals vastly improved but kept the gameplay same.

As for consoles, Chrono Trigger for SNES is the most remarkable game I can think of. It takes time travel very well, I was honestly surprised how such a game was made at that era.

If you want a more casual game, Puyo Puyo for Genesis could do well. It could be hard when you first get into it. The games back then were written to be more difficult to get more playtime out of games; and the AI can easily catch you off guard with big chains. (Its english counterpart for Mega Drive is called Dr. Robotnik's Mean Bean Machine, but the game is easily understood as it is already.)

The Legend of Zelda: A Link to the Past

Super Metroid

Donkey Kong Country 2: Diddy's Kong Quest

Chronotrigger

Advance Wars 2

Psychonauts (maybe that's too new idk)

Final fantasy 8. Squall is the best main character of the franchise. I love that game.

The Simpsons: Itchy & Scratchy in Miniature Golf Madness was a fun one I played growing up

Crash Bandicoot, Star Ocean 2, Legend of Legaia, Klonoa, Final Fantasy 7 of course, Kingdom Hearts 1

Earthbound, Super Mario World, Chrono Trigger, Donkey Kong Country, Sonic & Knuckles, Bust-a-Move, Vectorman

These are made for X86, so I'm not sure they technically count as "old games", but I've had a blast with recompiled+ported versions of TLoZ:OoT and Perfect Dark.
Ship Of Harkinian in particular adds so many features and improvements, that I'm not sure I would enjoy the original OoT played on an actual N64.

They may still hold up when emulated, though...

Not a childhood game, but the Japan-only SNES strategy RPG Fire Emblem: Genealogy of the Holy War is a great handheld emulator game! There's English patches for it.

Some random games from the mid 90s and earlier that I enjoyed when I was younger:

PS1: Final Fantasy Tactics, Final Fantasy 9, Lego Racing

SNES: Final Fantasy 4, Super Mario World

GBA (not of this era, but they'll probably run on your Anbernic): Fire Emblem (any of the GBA games, but Blazing Blade aka just Fire Emblem is a great starting point), Metroid Fusion

GBC: Link's Awakening DX

Edit: GB: Donkey Kong 94

MediEvil. The original, not the remake of course.

Grim Dawn, SMW, asteroids, metal slug, ms. pacman, centipede. Oh and defender and joust

Somehow my tastes stagnated and ossified so I mostly only play my childhood games. That said those that stood up the most are

• Super Smash Bros Melee
• Super Mario 64
• Wave race 64
• 1080 Snowboarding
• Rise of Nations

Those 5 games have cumulatively, for better or for worse, consumed a pretty sizable chunk of my lifespan. Sounds depressing to think of it that way, but then they did such a good job of being fun that I continue to enjoy that time so you got to hand it to them, they're hella good games.

• Shadow Man
• Turok 1 + 2
• Quake 2 (that OST kicks ass)
• Duke Nukem 3D
• Descent 3
• Commander Keen 4-6
• PowerSlide
• Ubisoft POD (Planet of Death)
• Terminal Velocity
• Tron 2.0

The biggest part of my childhood was Shadow Man. It was the first game I ever played that built a big world to explore with excellent narration and a good sense of progress. It felt rewarding to finally play all the way through to the end after many months. Some of the levels and music gave me nightmares as a child, but it was worth it.

I enjoyed a lot of story-driven games since then, but this one will always have a special place in my heart. It's so cool that NightDive remastered it recently.

Castlevania symphony of the night is a gem

Defender of the crown, c64 version.

Splinter cell: blacklist Metal gear solid

I recently put another dozen hours into SimCity 4.