Can I try a very brief (and not well-informed) attempt at an answer?
Before Buddhism there were the vedas. Atman is mentioned in the vedas but not well-defined.
After the vedas (and maybe still before Buddhism) are the upanishads. These sort of define Atman, by explaining that it means "soul" rather than "mind" or "body", and associate it with Brahman.
Then the Buddha comes along, and says, "Don't try to define what 'the self' means (that just leads to a thicket of views)", and let's have a "without-self" doctrine instead of doctrine about "self".
After the Buddha, Hinduism continues -- to some extent it's influenced by Buddhist doctrine -- see e.g. the Advaita Vedanta school's doctrine. I think it teaches non-duality (e.g. "Self and Brahman are One") but with some doctrine similar to Buddhism's (e.g. "the 'Self' is not the skandhas").
I think that's at least a little similar to Buddhism, in that one of its doctrines is "not this" (neti neti), i.e. "the 'Self' is not this" -- that seems to me similar to Buddhism's saying, "this is not Self".