Dragon Sorcerer Review

Gonna preface this by saying that it's gonna be more about design than balance.

I think that the level advancement for the dragon could be done a lot simpler; with different things at each level, it gets kind of noisy. Simplifying it to something like the way the Artificer's companion for that one Archetype it has does it would probably be best; too tired to remember the specifics atm, but I'm pretty sure it's akin to the way Cantrips level. This has the added benefit of making it so that the upgrades you get feel a lot more impactful, because you're giving out less of them. You should generally be taking into account the average levels players will run through for specific types of campaigns (I forget these numbers). It's also just very intimidating to players to wanna try out a new Class and see a behemoth of text.
I'm also not sure how I feel about your dragon's stats not being linked to yours; in my opinion, ideally, your spellcasting modifier (Charisma?) should represent the link/control/bond/etc. you have with your dragon companion.
I *really* like the fact that you can mix and match minor/major traits to get whatever kinda dragon you want, but I also feel like it could be done better; less traits make your choice in traits matter more. I'd personally go for something similar to the way that the Renegade Fighter (from that one League of Legends D&D adventure) handles its' firearm traits. At 3rd level (when you pick up the Archetype), you get to choose a minor and a major upgrade; at 5th, you get another minor; at 10th, you get another major.
I also feel like you've got too many features for a Full-Caster Class. As mentioned before, Spellcasting Classes get less Features, or less powerful Features, because a portion of their Class is defined by the fact that they get new spells when they level up, along with more Spell Slots, and other nonsense that Martials don't get.
Heart of the Dragon is neat, but it should probably make the Dragon resistant to that type as well.
I think Art of the Rider would work better as a separate Archetype. I'm still of the opinion that dragon-companion Archetypes make more sense than an entire Class, but if you're gonna go the Class route, it'd make sense to have dragon-riding be an entire Archetype due to its difference in styles.
Draconic Fortitude doesn't really make sense here since, the way it appears to be, you're supposed to cast while your dragon is the physical part. The situations where something like this would make sense would be, again, in a separate Archetype focused on mounted combat. Legendary Resistance pretty much nullifies the point of Draconic Fortitude, anyways.
Dragon's Assault is really strong. It basically doubles the damage-per-turn of your dragon.
Draconic Likeness is *also* more-so something that fits into an Archetype.
As a reminder, you have to watch out for and be really careful about the action economy. Tamer Classes aren't really a thing in 5e because it's hard to make it so that it's not automatically powerful as a result of having another action.
It feels like you have a couple different things competing for the same thematic space, which can be dealt with by turning them into Archetypes. Also, again, Full-Caster Classes get less Features than others.