FF X did. I still hear that opening theme and feel something. I doubt there will ever be a time in my life where I don't from that game. OOT was another.

FFX had absolutely stunning music.  

Shenmue 1

Yakuza 0 (Legit cried at the ending)

Nier Automata

Grandia

Tales of Vesperia

The Last Story

Xenoblade 1 and 3

Xenogears

Final Fantasy IX

Final Fantasy VII

RiME

Brothers a Tales of Two Sons

Lost in Shadow

Lost in Rain

Journey when I first played it.

Celeste.

Both Ori games.

Hellblade.

Gris.

Hollow Knight gets to me just due to scenario design and soundtrack.

Lowkey Uncharted 4 as well, as a fan of the series the ending concluding Nate's story got to me.

Several RPGs have heavily emotional moments, franchises like Final Fantasy and Tales for instance. 

Ico, SotC, Journey, The Last of Us, Syberia The World Before.

Perfect profile pic of a "little fuzzy". You ever play the Telltale game? I like it but it's not as good as the action game from 2021.

For me, some comes come to mind

-FF7 Rebirth: "No Promises to Keep" in the story and the credits in particular. 

-Animal Crossing series: The feeling they give me is so tough to describe but it's like going home.

-The Last of Us: The house argument AKA "you're not my daughter" and the "No Escape" scene moved me more than most scenes in gaming.

-The Walking Dead Telltale Series: Some of the key deaths and sacrifices really tug at the heartstrings.

-EarthBound/Mother series: So many emotional moments especially in 3 and the credits are moving as well. 

-Ocarina of Time: Fairy Fountain/Title Screen and the Temple of Time.

I had a couple like those yeah,

- The Walking Dead 1st Season ending is still amongst my top "gut-wrenching" moment written in fiction. There's always something about the cruelty of one's fate to be left alone due to unfair circumstances, especially for a child. Beautiful but incredibly sad nonetheless.

- Xenoblade Chronicles 2 and Xenoblade Chronicles 3 or even the whole main series in general, it's just about the result of someone who wanted to tell an ambitious tale of fiction in a fantasy sci-fi mix with the most incredibly human characters possible and the closest allegory to our own earthen experience.
The joyous highs, the discovery of life and it's cycles, death, it's relation to memories to what constructs us to who we are today. What our time means to us, etc ...

- FFX. I mean people pointed it out, but the moment you open the game and you hear "To Zarnakand" theme ... You can kinda know deep down where this is going. Honestly, one of my preferred tale of love in any JRPG.

- Kirby ... As a whole. Joy is contagious and Kirby will give that to you :)

Oh, Animal Crossing for me too. Lots of happy memories and a few sad ones. I remember feeling so heartbroken when a neighbor moved away in the GameCube original. 

Ocarina of time
Majoras mask
Journey
Xenoblade
Mario 64
Final fabtasy vii

It really takes you places, doesn't it? I played that whole game back in the day alone, every single night in the dark until it was finished. I got swept into the thing as if I were in the freakin' NeverEnding Story or something. I'll never forget it. It has a piece of my soul.