Final Fantasy 6 was the first game that ever made me feel an emotion. People give credit to 7 for that due to one major death, but the truth is 6 was a pre-apocalyptic and post-apocalyptic game. You honestly felt bad for the world seeing what it became after the villain effectively won the game.
The character cast was one of the best in a video game ever. The large main cast all got personal stories even if you had to hunt for some. Some had lost loves, murdered families and one even tries to commit suicide. Some had positive stories like adopting orphans or brotherly bonds. This is the first real feels in a game.