For my answer, I'm going to be focusing on the first 5 titles. That's because these were the games Sakaguchi directed before becoming a producer. I'm going to list some characteristics, ideas, and themes that, without which, a game would be hard to define as Final Fantasy.
1. Parties. Final Fantasy was not the first RPG on a computer. Far from it. But it was the first to place such an emphasis on a party of adventurers rather than a single hero. Characters who go off on their own are ripe targets.
2. World of Exploration. I don't know if a distinct world map is necessary, but Final Fantasy games need to be big enough to require exploration and discovery.
3. Crystals. Crystals are sources or vessels or whatnot of magical power. They are either valuable weapons for the heroes or need to be protected from evil. Classical elements are common, but not necessary.
4. Recurring Names. When a niche is open, fill it with existing mythology. The scientist is Cid, the dragon Bahamut, the fire creature Ifrit, etc.
5. Youth. Children. adolescents, and young adults are the best candidates for the job at hand. This matches the whole exploration thing pretty well. Older characters can be mentors and otherwise important, but it comes down to the youth to save the day.
6. Rebellion against Authority. Though not all authority is bad, the bad guys are either an empire, a dominating evil that rules the land, or some other authority. Thus, the player ends up being a freedom fighter.
7. The past. The world is in decline, or is built on the ruins of an earlier empire, or was built as a reaction to a past event. Somewhere in the story, some history will be necessary. This plays into the wise old mentors and exploring the world. The Younger playable characters will in a way be the first generation that can put the past behind them; or may simply be the last to live with that burden.
8. Gender. Although men end up being on the front line more often than women, each gender has a place in a party. Each gender may also express more masculine and feminine traits than usual. Once again, younger generations often have more sexual openness than older ones.
9. Finality. Games can be entirely self-contained. The evil must be completely eradicated, the heroes save the day, the world is at peace, no major loose ends. Sequels are permissible, but entirely unnecessary to complete the story or world.
10. Music. Although graphics are sometimes important, the graphics of Final Fantasy only became a chief concern around FF8 or so. Before then, they sufficed when necessary, but the job of creating mood and atmosphere fell upon the music, which digs deeper into peoples' heads and lasts better.
11. Roles. Every character has a role to play in the party. Even if they start with similar stats, different paths should be available so that they serve the parties in different ways. If two characters are functionally identical, either the game broke or you're doing some sort of White Mage Only run.
It is NOT:
1. A Complex Story. Final Fantasies 1, 3, 5, and many of the spinoffs have relatively simple stories. Until FF6, no game relied on its story in order to make the player care about the world. That's the job for exploration.
2. Graphics. Final Fantasy 13 had crazy graphics. FF4's sucked, even for its time. FF6 had alright graphics. All of these games had different art styles. None of them betrayed the Final Fantasy tradition.
3. Turn Based Combat. Final Fantasy has led active time JRPGs since FF4, and a majority of titles have used a form of this. As long as strategies and different options can be chosen in the time given, active time is A-Okay.
I'll add more if I think of them.