In our upcoming campaign starting at level 27, one of my players is going to play an Oracle 20/Paladin 7. Both the Enlightened Philosopher and Paladin provide a bonus equal to their Charisma to saving throws. My question is, do they stack?
No, they do not stack.
It is clarified in this FAQ that bonuses from an ability score are not untyped, but actually typed to that ability score. There are ways to add multiple scores to a single roll, but you can never apply the same ability score twice.
the paladin's untyped "bonus equal to her Charisma bonus (if any) on all saving throws" from divine grace is considered to be the same as "Charisma bonus (if any)", and the same would be true for any other untyped "bonus equal to her [ability score] bonus" constructions.