Dark KnightWe see the armor in two places. We can see Sidurgu sitting in the Forgotten Knight with the armor as soon as we enter Ishgard, however its entirely possible to avoid Sidurgu altogether and see it for the first time on your darkside alter-ego. Of course it means the armor is iconic but no indication that it helps master your darkside or anything of the sort (akin to how the AF1 warrior armor was meant to amplify the inner beast).

AstrologianThe astrologians of Sharlayan in the questline all wear the AF2, obviously it is typical gear.

Dragoonthere is no appearance or passing mention of the dragon lancer gear, however the pieces themselves follow german naming conventions (With the exception of the Mesail which is old french and plackart), the only other place that follows it were the name of the weapons in the Aery which is where you can find ancient Ishgardian armor pieces (the orthadox sets which the knights in the echo flashback wear and wield). 

This has led me to believe that perhaps this was the gear back when the Elezen and Dragons lived in PEACE. Dragoons in previous FF games, especially FF4 were known as dragon riders and the armor would not only make sitting on a dragon more convenient but also works with the name "Dragonlancer" and the design being more reverent or dragon-like in design. 

Haldrath wears the original AF1 Dragoon armor and so it shows the armor design is 1000 years old at least and most likely made specifically to kill the dragons.

This theory can only be overturned by the Kain pet, which states that Kain is a hero of old Ishgardian history who abandoned his post for honor, while a nod to FF4, the design is definitely reminiscent of the Dragonlancer gear which would date it as armor in circulation during Ishgard's early years. Or maybe, since Kain is known as a serial traitor in his games, the fact he would have the armor from the times when Dragons and Elezen were allies would make sense.

WarriorWhile I have not done the quests, I do know you dig up Mythril Heart's axe, I've never seen the axe but if it matches the current esoterics one then it makes sense that the armor is just another warrior gear of a particular mountain tribe or specifically Mythril Heart's set.

MonkI've seen screenshots of the MNK AF2 showing in the quests so I can only guess its just another temple sect uniform, please confirm.

PaladinPaladin armor as we know it has existed as far back as at least the thorne dynasty as evidenced by the life size displays during Little Ladies Day. So where the armor stems from is a mystery to me.