From a lore perspective this is definitely a possibility as it's kind of hinted at in the Dragoon 60 - 70 questline, however the war only just ended and I doubt all the dragons/dragoons who have lost loved ones are ready to forget all that just yet. I see dragon companionship as the eventual end game but it will take time and diplomacy.

From a gameplay perspective I'd like to see it happen but it would have to be different from the pet mechanics in FFXI, which were apparently pretty problematic. I'd like to see the pet as more of a temporary summon similar to how you build Nastrond that would buff your DPS but not be integral to your overall DPS. They could add another aoe jump to replace RoT, for example and give it a dragon-rider dive animation.