Cult of the Lamb, Fallout New Vegas, WoW, MGS 4, Bioshock 1-3, are a few that come to mind that move me in different ways (from making me care about the characters, to wanting me to see the deserved downfall of the antagonists, or caring about another character's wellbeing).

Cult of the lamb has me caring for my flock on a daily basis (I've married multiple followers, but I keep 2 distinct ones closer to me), keeping them well fed and breaking up fights between other members.

Fallout: New Vegas has me caring for different factions, but usually it's those within, like civs caught in a crossfire, or those unfortunate to what fate has dealt them.

WoW: generally heavily invested in the lore of that game. Despite having fought for the Horde for many years, I still held onto some compassion for the Alliance faction, so when I saw their world tree burning to a crisp, all the women & children that perished within it, I felt nothing but disgust, and a desire to see Sylvanas Windrunner & the Jailer pay for their crimes against life.

Metal Gear Solid 4, pretty much ended up caring about nearly everyone in that game, from Drebin being used as another cog in the machine just to live a life, to Ocelot having to live a completely fake life, just to finish a mission given to him decades prior. Snake I have always cared for, mainly because he was born/created as nothing more than an "anti-agent" against another man who wanted to enact Boss's dream and mantra (I also believe in the "world without borders" philosophy).

Bioshock 1-3 had me caring for what happened to the multiple diaries of other Rapture civs. I actually cared for the big Daddies, despite them opting to kill you if you ever tried to steal their little sisters, they were still once human and experiment against their will, so I still feel for them (not the splicers though, they act more like drug junkies and resorted to violence at will, as well as falling for indoctrination).

Xenogears again.

This event occurs in the past, during the era of Rene and Roni (who are the spiritual equivalents of Edgar and Sabin from FF6) about 500 years before the current time of the game. A war rages between Shevat and the Empire of Solaris, but also the rebel surface dwellers taking the brunt of the war on the side of Shevat (think of it like a Cold War, where the third world is attacked by Solaris). I see this whole part of the game's history as a rough retelling of the story of FF6, just instead of a mad clown Kefka, you have a devastated figure of the rebellion.

Solaris had with them an air fortress known as Merkava, which had the capability of a weapon of mass destruction (think Star Wars and the Death Star, or perhaps FF6's floating continent - something in the middle of those two things).

Sophia is one of the leaders of the rebellion - and controlled an airship from ancient times - The Excalibur - which she plans to kamikaze into Merkava, ending the war, and her life. Lacan is in communication with her, as she flies the Excalibur solo.

Sophia: "I'm going to end this here and now."

She accelerates the Excalibur, solo, toward its target.

Lacan: "Sophia! Don't be foolish! We'll help you escape. Please stop!"

Sophia: "People support each other to live, it's what brings us happiness."

Sophia: "So share that happiness"

Sophia: "And please... Lacan! live!"

and those were her final words as the Excalibur smashes into the Merkava, destroying it.

Lacan was devastated, and felt he was powerless to stop her death.

He went seeking something that would give him the ultimate power, the Zohar Monolith (known as the Conduit in Xenoblade) the Zohar, believed to be a connection to the divine - but was actually the power core of a star-killer created for a galactic scale war, crashing into the planet around 10,000 years earlier and taking over the evolution of life (much like Lavos in Chrono Trigger, and the Monolith from Arthur C Clarke's 2001).

Lacan also found ancient weapons from that time, the Diabolos (for Xenoblade fans, the Aions of Xenoblade Chronicles 2) which he used to wage a vengeance war against Solaris and Shevat, who he felt was responsible.
Sophia's last words ring in his mind: "Live!"
Lacan: "I will live..."
Lacan: "Even if I go to hell, I will live till the end of this world."
Lacan: "And if the world does not come to an end... I will destroy it with my own hands."

This event is kind of like when Kefka destroyed the world, because Lacan destroyed cities and killed roughly 98% of the world's population.

A bit of trivia:
Sophia is named for the Roman Gnostic Aeon Sophia (Roman Gnosticism is descended from Middle Platonism and in the same family of philosophy and theology as early Christianity and the neoplatonist movement).

Persona 3-5, Metal Gear Solid 1-4, Yakuza Games, The Last of Us.

Big emotions and iconic story moments with fantastic characters have greatly enriched my life.

Anime has perfected it, but video games can be very strong at it too :)

I'm playing cult of the lamb now and I can't care for these little pricks, one of them asked me could I cook some shit up for him and so I abided, I wanted to be a good cult leader. He gets sick, dies and I forget about the camp for a bit. Half them get sick, I'm down to 8 followers, 4 of whom are old. If I'm supposed to feel something for this little fucks I don't get why, half of the time I'd nearly sacrifice one to continue a run if I didn't need them to level up. One even asked me to murder him while he sleeps. I've had to murder to already who started spreading false lies among my followers about me and one little prick punched me in the face. I feel nothing for them asides as a means to level up. I suppose that's the intent though.