One of the new spells from Xanathar's Guide to Everything is dragon's breath (p. 154). Even though the spell seems like it involves attacking, nowhere in the spell description does it specifically say that you are attacking with it, only that you use an action.
Can a familiar that has the dragon's breath spell cast on it use its action to use this spell? Or would it not be a viable thing for the familiar because of dragon's breath technically being considered an attack?
Based on What counts as an attack?, I would assume that this would be a perfectly acceptable move, but again, not entirely sure.
Yes, the familiar can use the effect of dragon's breath
The Find Familiar spell description says "A familiar can’t attack, but it can take other actions as normal". The term "Attack" means you're making an attack roll. You don't need to attack in order to use the effect of the Dragon's Breath spell (XGtE, p. 154), because there is no attack roll involved, but a saving throw instead:
Per the rules on making an attack, for something to be considered "an attack" in 5e, it must involve an attack roll or otherwise must be called "an attack" explicitly in the rules (as with grappling and shoving):
See also: What counts as an attack?
Since the familiar isn't making an attack, it can use the effect of the Dragon's Breath spell.
Jeremy Crawford confirms this
As @V2Blast has pointed out, an unofficial tweet from rules designer Jeremy Crawford supports this answer: