Well, do remember that Ishgard and the whole of Coerthas in general is kind of on a witch hunt. Except it's a dragon hunt. Because they hate dragons.
Unless there's some sort of revelation late in the Dragoon job quests (there might be, I've only taken it to 34), it actually doesn't make much thematic sense for the lore's Ishgardian-based class to partner with dragons. That might just be a gap in quest knowledge on my part though.
That might just be somethng that is solved in the "see how the story progresses" content; maybe Dragons are a Beast Tribe sometime in the future, who knows?