My attachment to this game is unhealthily strong, it was my childhood passion and I've always felt a really strong connection with it. I love pretty much everything about it but the feeling it gives me is the reason why it's my favourite game. No other game will ever be able to dethrone it.
But to give more solid reasons as well as that: I love FF8's world, it's one of the few game worlds I've played in that I can actually picture people living in. It's an awesome blend of retro, modern and futuristic themes that works far too well and it has some really nice locations. The Junction system is marmite but I love it, something about it has always felt really engaging to me, and the feeling of power you get when you learn how to exploit it is awesome. The music IMO is the best in the series, but that might be because of my attachment to it. It has the best summons system in the series, as it makes them feel almost like minor characters rather than just spells for you to cast, and I really like the way it works together with the Junction system. Though I guess FFX does a similar thing. I love the visuals, at the time the cutscenes were mindblowing, and because of my childhood love for it I really love the graphic style it and Parasite Eve have, and I love the old prerendered environments too.
As for the story and characters, admittedly FF8 is pretty lacking in the character department for the most part, most of the party members are just kind of there, but the story that FF8 actually has is much better than people give it credit for, it's just very poorly told. The most important plot details are hinted at really subtly instead of actually made clear to the player. I'm considering making a YouTube video or something about it someday exposing its 'hidden' story for everyone to see. Ultimecia is a pretty interesting and sympathetic villain when you understand what's going on.