Dragonborn are humanoids
This isn't stated with very much clarity in the Player's Handbook. Possibly the most direct statement is from page 11, the first paragraph under "1. Choose A Race":
Every character belongs to a race, one of the many intelligent humanoid species in the D&D world.
So all the player races are 'humanoid' type. Any deviation from that would be handled by a specific racial trait that says what type, types, or tags they have.
Subtypes aren't a thing anymore; what you are referring to are technically called 'tags' (as described in the Monster Manual, page 7), like elf, halfling, lizardfolk, dragonborn, or shapechanger. Humanoid races are usually listed in monster stat blocks with a tag indicating their specific race, so a dragonborn would be a "Humanoid (dragonborn)", but nothing more than that. There's no 'reptilian' subtype for lizardfolk, for example, and dragonborn are certainly not dragons.