The game OneShot was one of the few to ever have me emotionally invested to the point of tearing up. An excellent aspect of that game is having you the player be a character in the story in a organic way that makes sense plot wise. you converse with the character you are controlling, essentially a lost child that wakes up in a nearly dead world and wants to go home. You directly communicate, console, and encourage them. and it made me so much more invested. The original ending made you pick a very tough choice that leaves you emotionally deviststed, its dlc solstice finally gave a true definitive ending.
Everhood also gave me some emotions. Death is a very hard thing to contemplate and deal with, for me anyways. It helped make me a little more comfortable the nature of death and why its such an important aspect of existence. The game turns from fun cute psychadelic rythm game to heavy existential mercy killing of immortal beings who have gone fucking nuts and mentally decayed after trapping themselves in a legitimate eternity of immortality, and then their reality. The music is kick ass and the gameplay turns the old rhythm formula on its head.