It's usually best to use the most recent printing of anything, and that means using the Spell Compendium version the 7th-level Sor/Wiz spell animate breath [trans] (SpC 11):
For this spell to function, you must have a breath weapon, either as a supernatural ability or as the result of casting a spell such as dragon breath (page 73). When you successfully cast this spell, you imbue the energy of your breath weapon with coherence, mobility, and a semblance of life [for 1 round/level].
Emphasis mine, and I'll address that shortly. One should also use the Races of the Dragon version of the feat Draconic Breath (102):
As a standard action, you can convert an arcane spell slot into a breath weapon. The breath weapon is a 30-foot cone (cold or fire) or a 60-foot line (acid or electricity) that deals 2d6 points of damage per level of the spell slot you expended to create the effect. Any creature in the area can make a Reflex save (DC 10 + level of the spell used + your Cha modifier) for half damage. This is a supernatural ability.
Emphasis mine, and there are two ways of reading that emphasized text.
The creature takes a standard action to convert an arcane spell slot into a breath weapon and uses that breath weapon as part of the same standard action used to convert the slot.
This reading means the breath weapon the feat grants can't be targeted by the spell animate breath. The spell slot is converted into a breath weapon that's instantly used, and, prior to that conversion (if the creature has no other breath weapon) the creature has no breath weapon to target with the spell animate breath. I assume this is the feat's typical reading.
Hence a caster could no more cast the spell animate breath on this reading of the breath weapon granted by the feat Draconic Breath than he could cast animate breath on the breath weapon generated by the 4th-level Sor/Wiz spell firestride exhalation [conj/evoc] (Dragon Magic 67).
The creature takes a standard action to convert an arcane spell slot into a breath weapon, and, afterward, the creature can take a standard action to employ the breath weapon.
This reading is extremely legalistic, and it's unlikely this was feat's intent, but this reading does allow the spell animate breath to be cast on the feat Draconic Breath's breath weapon (even if the breath weapon is made available through expending a 0-level spell). This reading also removes the utility value (such as it is) of being able to spend spell slots to make breath weapons that hurt foes right now. (The creature is essentially forced to take two rounds to use its breath weapon. And, no, that's not a delay expressed in rounds.)
Under such a reading the DM must determine if the creature possessing the feat can either take only one standard action to convert one spell slot into one breath weapon or take several standard actions to convert several spell slots into several distinct breath weapons. Essentially, the DM must determine if the feat is a switch or a dial. (This DM recommends a dial.)
The spell animate breath fails if the caster lacks a breath weapon
Although the spell's duration is 1 round/level, the spell animate breath nonetheless requires the caster to cast it on the caster's breath weapon. Otherwise, the spell fails. The typical reading of the Draconic Breath feat has the feat supplying a momentary breath weapon that's used in the same round upon exchanging the spell slot for the breath weapon, leaving no time for the spell animate breath to be cast.
A spell like the 4th-level Sor/Wiz spell dragon breath [evoc] (SpC 73), with its 1 round/level duration, is a valid target of the spell animate breath, the spell dragon breath even stating that, after the spell's cast, the caster must take a standard action to use the breath weapon and then must wait 1d4 rounds before using the breath weapon again. Until the duration of the spell dragon breath expires, the caster has a for-reals honest-to-Pelor breath weapon, not a fleeting one like that afforded by the typical reading of the feat Draconic Breath.
Question: Does the prestige class dragonheart mage (Races of the Dragon 88-91) change the statistics of the variant Huge fire elemental created by the spell animate breath?
Answer: No. The spell animate breath determines the variant Huge fire elemental's statistics, not the breath weapon employed to create it except insofar as dictated by the spell. Note: The prestige class dragonheart mage is, frankly, less than stellar, so allowing minor changes to the variant Huge fire elemental as a house rule shouldn't upset game balance significantly.
Question: Can the variant Huge fire elemental created by the spell animate breath be created anywhere within what would otherwise be the breath weapon's typical area or must the creature be created somewhere else?
Answer: Absent a range, the most conservative house rule for determining where the spell's effect occurs is adjacent to the caster's space in an area sufficient to accommodate the newly created creature. Note: Using such a house rule, a caster that lacks sufficient space adjacent to him will see the spell's effect fail. The created creature won't, for example, break down walls purely as a result of its creation.
Note: A Huge elemental can be summoned using the 7th-level Sor/Wiz spell summon monster VII [conj] (PH 287). While the spell animate breath has only somatic components and a casting time of 1 standard action, summon monster spells—because of their versatility and splatbook support—are usually far better choices for a sorcerer.
Yes, both effects can apply to the same spell.
There's nothing in the wording of either feature to indicate that it excludes other features adding damage as well. Many other combinations of features can work this way -- a barbarian's damage bonus from Rage is cumulative with their Strength damage bonus and other bonuses such as the Dueling fighting style. The word all of these have in common is add.
Contrast this with Thirsting Blade, which says:
You can attack with your pact weapon twice, instead of once, whenever you take the Attack action on your turn.
It doesn't say "you can attack one additional time", so the reason it doesn't stack with Extra Attacks is that neither one adds attacks, they both just change the number of attacks from one to two.
This is also similar to how the barbarian and monk Unarmored Defense features don't "stack" with armor. Armor provides one way of calculating AC, Unarmored Defense provides another. A given character can only use one.
So, in general, effects or features that let you add something to a roll are cumulative, but effects or features that change a value are not.
Best Answer
You're correct, Elemental Affinity applies specifically to spells.
This is probably just an oversight on the part of whoever wrote the character sheet.
With that said, the DM is free to rule that this works, and it's a pretty cool and reasonable idea. I'd certainly allow it if one of my players took the same dragon type for both their Dragonborn race and their Draconic Sorcerer.