We know that a familiar can use Dragon's Breath establishing it is not an attack, nor is it a spell being cast (from this tweet). Does that mean a familiar under the effects of Sanctuary could use Dragon's Breath without breaking its effect?
This was changed by errata, so dragon's breath's action does break sanctuary
As of the November 2018 Player's Handbook errata, sanctuary ends when:
the warded creature makes an attack, casts a spell that affects an enemy, or deals damage to another creature
Hence, using the action granted by dragon's breath would end sanctuary as it deals damage to another creature.
RAW you are correct.
However, Jeremy Crawford disagreed on Twitter in a ruling related to a spell with similar wording to dragon's breath for dealing damage:
RAW: witch bolt exploits a loophole in sanctuary. RAI: witch bolt's damage ends sanctuary.
Casting witch bolt and then having sanctuary cast on you would make it harder for enemies to target you while still allowing you to do damage to enemies without ending sanctuary.
The dragon's breath spell would exploit a similar loophole. Crawford has explained that this is not the intent.