Atman is not the personality, the body, or the stream of thoughts. The Upanishads describe it through negation: not this, not this — Neti-neti. What remains when everything that can be observed is set aside as not the observer is Atman.
Advaita Vedanta‘s central teaching — tat tvam asi, “thou art that” — is the identification of this Atman with Brahman, the one without a second. Buddhism diverges: Anatta denies that any such permanent self can be found. Whether the two positions finally disagree depends on what one means by “self.”