Nostalgia is a very incomplete explanation, kinda fallacious, even. I recognice that I cannot speak objectively about this game, for it was indeed my first foray into the Final Fantasy saga, so it would be better explained by a different perspective.
Many years after its debut, about 13 years later, my young cousins had their first contact with the ffVII universe. Unlike myself, they didn't had any rose tinted glasses, mind you. They grew up the PS2, ffX was their first RPG, they were already moving into the PS3 and simply couldn't understand what the fuss was all about since ff VII had butt-ugly graphics and a "nosensical waiting to hit each other" combat system. I explained that it had to do with the plot and characters, and they of course didn't buy it.
Until they stumbled with a copy of Kingdom Hearts. They inmediately fell in love with the Sephirot and Cloud characters, hard. "The coolest characters in the entire game, better than Sora!" they claimed, and needed to know more about them. Out of their own volition, they bought Crisis Core asap for their PSPs in order to get their ffVII fix, and were blown away by its world and mythos. After that, they really, really needed to play the original and see the origins of Sephiroth and what the hell happened to Cloud.
Thing is, ffVII is massively appealing to teenagers, regardless of whetever they grew up with this game or not. The game oozes coolness (longass swords! trenchcoats! beautiful warriors as love interests!) the main character's struggles have to do with forming their identity and distinguish what is real from what is made up (a pretty existential issue for any teenager) and its mood swings wildly between brooding drama and ridiculous, unpretentious cheesefest (kinda like teenagers theirselves). So of course this game is going to impact whoever is in their formation years, which is different from merely "nostalgia" per se.
In addition to that, and as a videogame critic, the game was a prime example of videogame narrative done right due to its nature as a transitional game between the more artisanal, personal videogame making of the 16 bit generation, and the blockbuster nature of the massive AA titles that would define the next generations to come. Yes, it had cutscenes and bombastic production values (for their time), but the game truthly shone when the less typical narration kicked in. Playable flasbacks. Showing the strenght of your antagonist trought combat, not cutscenes. Player-controlled love triangle. Cross-dressing sections. And a long etc of other neat, experimental narration tricks impossible to replicate in any other medium. Now, mix it with a hefty, crazy mix of wildly politically incorrect elements (terrorists as heroes, transexuality, etc) you get the kind of game that would be almost impossible to get greenlighted today. And yet here we are, celebrating its remake. Incredible.