Ooh! Nostalgia time!
I have very vivid memories of Sonic, Pac-Man 2, Moto Racer 97, Contra, Final Fight 2 & 3, Knights of the Round, Earthworm Jim, Spyro, Phantasy Star IV, and Mega-Man. I don't remember a lick of storyline but when I close my eyes, they're there.
As for games that have a more emotional tether, that would be things like Legend of Mana, Chrono Cross, and Threads of Fate. All of which helped build my love for fantasy. I was enamored by the art style and gameplay of LoM, the fact that you could mix and match different skills and discover new ones was like a drug, add in a crafting and pet system and I was stuck. I have a deep love for the soundtrack of Chrono Cross, the combat system also challenged my stupid child brain. Threads of Fate was more of a casual playthrough, but I remember going back to it over and over again, because I kept losing my saves.
I also have games that were more communal: sometimes played with cousins and siblings or with me as the riveted audience. Things like Bomber-man, Tetris, Puyo-Puyo 2, Worms, Resident Evil, Final Fantasy 7-9, Valkyrie Profile, Prince of Persia and Grandia. I have stories for all of those but this is already getting too long. 😅