The first time each turn the affected target takes damage of the chosen type, the target takes an extra 2d6 damage of that type.
Would this damage be considered to by made by the Elemental Bane spell and its caster?
For a scenario where this might matter, consider the Awakened Spellbook ability from the Order of Scribe:
you can temporarily replace its damage type with a type that appears in another spell in your spellbook
So if the additional damage is considered to be damage dealt from Elemental Bane, you could change its type. As a result, Elemental Bane could trigger from a fire hit but deal piercing damage (inspired by this question).
This is just an example of why this would matter and there might be other features interacting with the source of damage eg the Pyromancer Sorcerer's Fiery Soul that ignores resistance for fire damage.