Huh, it absolutely does. Alan Macdonald wrote a book about geometric calculus and made a companion video series for it. Here’s the section defining manifolds: https://www.youtube.com/watch?v=jfwdlW7Yr_I

That said, tensors are in fact more general than geometric algebra’s multivectors (in that the latter form a quotient algebra of the former). But not all tensors correspond to manifolds as we know them in physics and I argue that GA keeps track of the geometric properties we want better.