I was going to say Rayman Advance for the Gameboy Advance, but that was my first game as well as my first unfinished game...
I was also going to say Rayman 3 for the Gameboy Advance, but despite completing that game 100%, I don't think that solidify myself as a gamer to me...
So, I'll just say that Tales of Symphonia for the GameCube was the game that made me a gamer, which is funny to me because at first, I was only playing this game out of spite for someone else who would not shut up about how awesome the game was... But seriously, it was the first game I ever invested most time in comparison to the other games I've had beforehand, the first game (with an actual end to speak of) to be played more times than I could count at this point, and the first game that I ever went in deep discussions over with others who have played this game as well...
Granted, this wasn't my first [J]RPG (especially if you're counting Pokemon Ruby, Mario & Luigi: Superstar Saga, or even the NES Zelda games for the Gameboy Advance) at the time, but this was the game that kept me up late at nights, leveling up my team, experimenting with their EX skills, doing various sidequest because my curiosity got the better of me most of the time, and other random stuff for shits and giggles, basically... I played Symphonia more times and with more hours clocked in overall than Pokemon Ruby, which was the most social game I was playing among friends beforehand...
Overall, that feeling of playing something for hours on end, actually discussing not only the gameplay, but the story, characters, and sometimes themes with friends, and even finding out about others games in relation to having that feeling overall is why I consider Tales of Symphonia for the GameCube the game that made me a gamer...