One theory proposed by another in my FC is that Shiva does actually lord over the dragons, and hasn't tempered Ishgard, but has still manipulated them into their continuous Holy War to feed off the aether released by the massive loss of life from both sides. In this way, Ishgardians would have no reason to exhibit fanatical devotion to her by name, and in a way the dragons sort of do (I don't think we've encountered an actual dragon talking yet? Though there is the heretic "inspector", so I'm not sure where to go with that).

I could see several of these scenarios tossed out there as plausible and can't really choose one as a clear frontrunner in my mind, but I do think it's pretty obvious now at least that Shiva and Ishgard's stories intersect each other in some way, and that Shiva is most likely very subtle and underhanded in her dealings compared to the brash in-your-face personalities of Garuda and Ifrit.