from what I have seen it seems that she is the giant crystal in the flow of aether below Eorzea, only by our contact have we been given the power of the echo and the immunity to tempering (remember this happens at the start of the story, even 1.0 from the "hear, feel, think, which is Hydaelyn" But I don't think she is a primal as we know them, the closest thing we worship to the primals is the twelve, as they can be summoned, and given at least a partial corporeal form as seen in the end of an era, and was a big part of the end of the 1.0 story. It's also touched upon in the story when some from Little Ala Mhigo wanted to summon Rhalgr that if completely summoned the twelve could even posses the power to temper. So yeah I don't think Hydaelyn is a primal as we know them but perhaps something more than them.