Anatta stands in the most direct possible opposition to personal identity as substance and to any doctrine of an enduring atman or purusha, and the philosophical dispute between Buddhist no-self teaching and Vedantic or Samkhya accounts of a persisting self is one of the oldest and most carefully argued disagreements in the history of Indian philosophy, conducted across many centuries of formal debate between Buddhist and Brahmanical schools, each side developing increasingly sophisticated arguments against the other's account of what, if anything, persists through a person's life and across successive lives.
Advaita Vedanta's response is characteristically distinctive: rather than simply asserting a substantial self against Buddhist denial, Advaita argues that what anatta correctly identifies as lacking a fixed nature is precisely the empirical, individual ego, the very thing Buddhism analyzes and finds empty, while a deeper, non-individual consciousness, ultimately identical with Brahman itself, remains untouched by the analysis and is exactly what liberation reveals, a response Buddhist philosophers have in turn argued smuggles back in, at a deeper level, the same fixed essence anatta was meant to rule out entirely.
Personal identity as continuity, developed independently within Western philosophy by Locke and Parfit, arrives at a structurally similar picture to anatta, a person as a connected stream of psychological states rather than a persisting substance, without either tradition directly influencing the other, another case of the same structural claim surfacing twice.