I have a similar experience with my English learning journey, but instead of videogames, mine was with music.
I had started to learn English on a little institute in my hometown, but the real learning came when I became obsessed with a rock band when I was a teen, and I used to translate the songs with the little cd booklet and a very crappy english dictionary.
I would then go to the institute and ask my teachers if the translations were correct or not, hehe.
After that, I just took off by myself, and only went back to an institute to teach English hahahah.