#You can use it. Once.
You can use it. Once.
It's a spell, and it does deal damage of the chosen type (assuming you choose the same type for the dragon's breath as you have for your bloodline). That meets the requirements for applying the class feature to that casting of the spell. The spell likewise has associated damage rolls, and thus the bonus can be applied. There is no listed requirement that anything be a "direct consequence" of anything. It's just as applicable here as it would be with wall of fire or produce flame - each of which applies damage at times and based on triggers other than the instant of the casting.
This, as stated in the class feature, allows you to add your charisma modifier to one damage roll of the spell - in this case one exhalation. Any other exhalation would be a different damage roll, and wouldn't get the benefit. Timing here could get kind of odd, amusingly, as it never tells you you have to add it to the first damage roll of the spell, and it doesn't tell you if you decide whether or to add it before or after you roll the dice, but the "Yes it works, use it once." part is pretty clear.
The "one damage roll per spell" is accurate, though was not in the original printing. It's an errata.
Per The Player's Handbook Errata,
Elemental Affinity (p. 102). The damage bonus applies to one damage roll of a spell, not multiple rolls.