It's probably been stated already, but OP probably has never played a single Final Fantasy game in his/her life.

Traditionally, in FF game and lore, Dragoons are never Dragon Tamers, but Dragon Slayers. Classic Dragoons across the series, as far as I remember:
Ricard Highwind (FFII)
Kain Highwind (FFIV)
Freya Crescent (FFIX)
Oerba Yun Fang (FFXIII - Highwind is pretty much Jump)
Lancer (FF Tactics)

They all have one skill in common: Jump. As mentioned previously too, FFTA Dragoons had Elemental Breaths. But Elemental Breaths added a dimension of tactics for Dragoons, to exploit enemies' elemental weaknesses - in a game like FFXIV where elemental effects aren't as significant (Bombs taking damage from Fire spells or skills, for example), it is not really needed.

I don't blame the OP for saying Jump in this game doesn't really make him/her feel special, because in this game, it's exactly that - it does not add anything extra to gameplay, other than being an off-GCD skill that can be used to increase DPS whenever it's up.

There have been many suggestions to make Jump feel more..."Jump-y", for lack of a better word.
- Immunity to attacks/AoE while the Jump animation is in motion. This is especially because any of the previous characters, when they use Jump and are in the air, become untargetable, or immune to attacks that include them as an attack target.
- Larger damage. Could probably be made conditional so as not to make combat feel slower - Fang's Highwind Limit Break does some crazy burst damage when the enemy is staggered, so perhaps there could be some conditional debuff that stacks up to a certain limit, and makes Jump attacks do more damage based on the number of stacks, as well as probably consuming all of them after the use of Jump.

Other than FFXI, which I did not play, Dragoons in traditional FF never had dragon pets. They were spear wielding, jumping badass characters wearing signature dragoon armor, that brought a world of hurt to their enemies - especially dragon enemies. I think my only complaint about Dragoons in this game, in comparison to other games, are that Dragoons aren't as tanky as I'd like them to be. Although if they were made any tankier, tanks would be out of a job.