Vedrfolnir (one of Hrasvelger's children) reveals that knights flying on dragons were a thing before the Ishgardians and the Dragons had their falling out. This begs the question of what they were fighting at the time that required Men and Dragon to work together. It's highly likely that dragoons pre-dated the Dragonsong war and the Ishgardians just kept doing it given how effective they were.

Also, if you go far enough in the Churning Mists sidquest, you eventually get a series of quests that involve the other dragoons. They have the same armor/weapons that you do so presumably, they are all using/wearing arms and armor made out of dragon bones and blood. I don't see anything that would indicate that's special to just the Azure Dragoon except that they "earn" it through combat.