Question about this feature of the Sorcerer Draconic Bloodline: "whenever you make a Charisma check when interacting with dragons, your proficiency bonus is doubled if it applies to the check."

Would this feature apply to a creature of the "Dragon" creature type but that isn't a "Dragon", like a Drake, Dragonborn or a Wyvern?

 My assumption is that it applies to those with the Dragon creature type. Which doesn't include Dragonborn, by the way, since they are Humanoid.

Drakes and Wyverns are technically dragons, but Dragonborn are not.

Yeah, Dragon creature type (wyvern, drake, dragon turtle), but not dragon-like creatures like dragonborn and koblods.

It applies to any dragon type, not just true dragons but those  distantly related. Here's the description;

Dragons: Dragons are large reptilian creatures of ancient origin and tremendous power. True dragons, including the good metallic dragons and the evil chromatic dragons, are highly intelligent and have innate magic. Also in this category are creatures distantly related to true dragons, but less powerful, less intelligent, and less magical, such as wyverns and pseudodragons.

The Dragonslayer weapon is a prime exemple;

DRAGON SLAYER
Weapon (any sword), rare
You gain a +1 bonus to attack and damage rolls made with this magic weapon.When you hit a dragon with this weapon, the dragon takes an extra 3d6 damage of the weapon's type. For the purpose of this weapon, "dragon" refers to any creature with the dragon type, including dragon turtles and wyverns.