A couple games are 100% guaranteed to produce tears every time: Journey, and Undertale. Journey is a really beautiful and unique game, and manages to make me feel very deep emotions with absolutely no dialogue, other than "chirping" at other players you may run into during your journey. Undertale... god, so good.
Undertale. It's my favorite game ever. I almost cried while watching the closing credits at the pacifist ending
Almost? I must be very weak then because I am a sobbing mess every single time. That, and as soon as the first note hits in this part...
I am a weak little baby and cannot control myself
One game that really affected me as a kid was Legend of Zelda: Majora's Mask. Those were turbulent times, and something about that game was very calming to me. I was so invested in the individual stories of all the people I was helping, and the whole "saving the world from destruction via very angry Moon Man" was appealing to me at the time. I felt I was making a difference in the lives of these computer generated people.
Here's a big one: Animal Crossing New Leaf
This game came out during a time when I was very alone, and dealing with many brand new symptoms of mental illness which had me feeling very confused and afraid. This game carried me through much of that, it was very comforting to be able to visit this town full of cute little animals that care about me and are always excited to see me. The atmosphere in the game is so serene and akin to being wrapped up in a warm blanket in front of the fireplace on a rainy night. Unfortunately, I was a budding alcoholic and sold all of my video games for booze money. I have since bought the game again, and have maintained the same town for two years now.
Stardew valley sorta pulled on the same strings in me, and I did play it obsessively for a while.
I could ramble on forever on the subject, I get very emotionally attached to video game characters. I can't really get as immersed as I used to, but it's still an escape I enjoy from time to time.