As you say, they have the starting point of "This game has to be a multi-game, multi-media genre powerhouse," and they develop from there. They don't really have a clear idea of what they want the series to be, though, which results in the flavor of the year gameplay the series flits around rather than having any gameplay style it can call its own. They had the ATB with various twists for a while, but then they dumped it for a standard turn based system with rock/paper/scissors, which they then traded for MMO-lite, and then the nonsense paradigm-shifting system, and now we're an action RPG.

For what its worth, I appreciate Persona more these days, because they seem to have a more stable identity with consistent gameplay that the series builds on rather than flipping the tea table every gen. They also have the same philosophy of milking their properties dry with anime, spinoffs, etc, but I feel like they do it from a firmer foundation of identity then Final Fantasy has had for a while.