As far as video games go, the music at Grandma's house in LoZ: The Wind Waker after Aryll was taken really got to me. So did the Arena Champion's "Suicide" in Oblivion. (Especially because everyone thought I was an asshole for killing him afterward) Some parts of LoZ: Majora's Mask also made me sad. Silent Hill 2, of course.
Aaand... Let's see... Oh, Ty the Tasmanian Tiger was a definite one. Not because of anything that was actually in the game, but I played it during a time when I was going through a bit of an existential crisis, where I had horrible phobia-like terrors about the afterlife or possible lack thereof, (I was a weird little 8 year old...) to the point where I was almost in tears whenever I thought about it long enough and constantly resolving to spend more time with my parents and friends and other things that people resolve to do when they're actually on the brink of death. I associated Ty with those thoughts for a long time, to the point where the theme song that played on the outback crab-car sessions would invoke feelings of despair and loathing. Various things changed, and I got better, but man, I had some bad, bad times.
The endings of Dark Souls and Pathologic depressed me to no end, both because they just made me feel so hopeless, and the latter because it meant all of the characters, some of which I had grown to care about (GODDAMMIT, DO NOT HIGHLIGHT THIS UNTIL YOU HAVE REACHED AT LEAST ONE OF THE ENDINGS YOU MONGREL!) were quite literally just puppets in a video game, in their own canon!
But most of those made me sad last year, if not earlier. (Several years earlier in the case of the first two paragraphs.)
The two more recent sad gaming experiences I've had were in Presentable Liberty and Bloodborne.
Presentable liberty was a sad game, but Bloodborne had one sidequest where you had to help a little girl find her mother.
Basically, her mother went out to find her father, a hunter who got so bloodthirsty over the years that he turns into one of the monsters himself. But her mother forgot to bring the music box that reminds him of his human self whenever he turns. Needless to say, you're forced to kill him, and her mother is fucking dead. When you break the news to her, she's heartbroken, and she begs you not to leave her alone, but... You have to! You have to kill all the monsters on the hunt and save Yarnham until next year...
And no matter what you do, no matter which sanctuary you do (or don't) send her to, when you're away on the hunt, she leaves the house, loses her way... And you find a little red ribbon in the guts of one of the nearby monster pigs.
I don't know why I sympathized with her more than other sad little characters in other video games, but that one made me hurt inside.