It's open to interpretation.
However, it does allow your spells to do so, in conjunction with your 6th level feature, Fire in the Veins:
At 6th level, you gain resistance to fire damage. In addition, spells you cast ignore resistance to fire damage.
(Emphasis mine.) Since this is a separate feature, there is no question that this can be applied to the 'result' of the Fiery Soul feature. As you quoted, the feature:
treats immunity to fire damage as resistance
Since Fire in the Veins allows the Pyromancer to ignore resistance to fire damage with its spells, it will deal full damage to a creature with resistance or immunity to fire damage—but this is only definite with its spells.
As for whether the feature 'stacks with itself', this is a matter which will come down to the DM you're playing with, regardless of your reading. Linguistically, you ignore resistance [to fire damage] and treat immunity as resistance. There's no 'instead' or 'alternatively', so if you read the rules literally, it seems the answer to your question is yes.
One final thing to note is that this feature leaves no space for choice and makes no distinctions: if you take the above interpretation, you will also ignore your own resistance and immunity to fire damage—even that provided by the same feature.