There isn't necessarily a stack with call/cc, the model is that activation frames are heap allocated[1] and can be arbitrary composed. Normally they would be GC collected when a function returns, but call/cc exposes the caller activation frame (+ a function representing the rest of the caller function body) as a first call object, that once captured prevents it from being garbage collected. The stack is recovered by realizing that activation frames have a reference to the calling function activation frame (which is passed to every function as an implicit parameter).

Also an activation frame per se is immutable, but not the objects referenced by it, so all modifications are preserved when invoking a continuation.

[1] this is just a model, in practice there are many ways to implement this that optimize for the common FIFO allocation discipline.