How to Play: How to Train your Dragon!
Second place in the poll was HTTYD! There's, in my opinion, two distinct appeals about dragon riding settings: High Flying combat, and Raising an extremely powerful buddy that's bonded to you, and we'll go over both! Unlike Magical girls, dragon riding isn't actually as popular in TTRPG spaces, so we'll need to go over games that mechanically fit but don't fit flavor wise, but I'm sure most people are used to re flavoring games here, right?
Flying Circus Good for: Tactical, customizable flying action Flying Circus is already a fantasy game, and you could totally reflavor some of its mechanics around dragons, the primary issue is mechanics focused around the more mechanical part of planes, but you could flavor that as creatures being temperamental or rowdy
Beam Saber Good for: Faction drama, Vehicular combat Beam Saber's whole gimmick is that its focused around Vehicular combat, you could very easily reflavor vehicles as just different critters, the primary problem would likely come from reflavoring Beam Saber's sci-fi stuff, but A: i'm sure some of you want dragon riding sci-fi bullshit B: "Any sufficiently advanced technology is indistinguishable from magic" goes both ways
Animon Stories Good for: Creature bonding stories, Focusing on raising and fighting Animon is a digimon game, but with some minor tweaks you could play it as a creature riding game, I mean really all you'd need to do IS to just make it so riding creatures is a core part of the game, beyond that the game is already pretty suited for this style of play
Warrenguard Good for: Basically everything Warrenguard is your go-to, its explicitly a game about dragon riding and it wears its inspirations on its sleeve, its gameplay is deeply connectect to both raising a powerful creature and high flying questing. it's also just a good game!