The Katha Upanishad is framed as a dialogue between a young seeker after the truth of what happens to the person after death, and the Lord of Death, Yama. This dialogue sets forth the perspective of the Vedantic approach.
Sri Aurobindo clarifies the vedantic approach: “he admits an identical, a self, a persistent immutable reality,-but other than my personality, other than this composite which I call myself.” Vedanta frames the question of the survival of the personality thus: “…even the gods debated this of old and it is not easy to know, for subtle is the law of it; something survives that appears to be the same person, that descends into hell, that ascends into heaven, that returns upon the earth with a new body, but is it really the same person that thus survives? Can we really say of the man “He still is,” or must we not rather say “This he no longer is”?
When the Lord of Death responds to Nachiketas, “Yama too in his answer speaks not at all of the survival of death, and he only gives a verse or two to a bare description of that constant rebirth which all serious thinkers admitted as a universally acknowledged truth. What he speaks of is the Self, the real Man, the Lord of all these changing appearances; without the knowledge of that Self the survival of the personality is not immortal life but a constant passing from death to death; he only who goes beyond personality to the real Person becomes the Immortal. Till then a man seems indeed to be born again and again by the force of his knowledge and works, name succeeds to name, form gives place to form, but there is no immortality.”
Clearly the Vedantic view acknowledges something which persists, but this is not something tied, as in the popular conception, to a specific personality-entity. Rather, there is a Self which puts on the various bodies and personalities, gains the essence of the experiences in each lifetime and uses these to mature and grow.