To my knowledge, each character can only cast 1 spell during another character's turn.

my question is - can a character who has Ancient Names (or a Genie for that matter) cast all the spells on the card during another character's turn, or are the spells on that card bound by that rule as well?

We had an argument about it, since basically it is said that the spells are not actually being cast by the character, but by the card, so we're not sure if the card is bound by the same rules as regular characters.

The Genie is covered by the current FAQ 1,1, so you can apply the same answer to the Spells on Ancient Names Quest Reward. Check the FAQ on page 7:

Q6: Do Spells cast by the Genie count against a character’s Spell limit?
A: No. Note that the phrase “treat the Spells as though you had cast them” is only taken into account if an effect refers to a character who cast the Spell (such as the Reflection Spell).

This answer may not be 100% clear, but the meaning is that Spells on those cards don't count as your Spells, so they don't count as Spells cast by your character except for the purpose of Spells such as Reflection or Hydra Spell. They are treated separately.

Yep. Genie spells no county to spell limity