In physics, a field is a physical quantity, represented by a scalar, vector, or tensor, that has a value for each point in space and time
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point.
Reading these definitions, I’m confused. Is it possible to say that a manifold is also a field? (Because both are mathematical objects with a value at each point.)
What physicists all a "field" people in DG call a "tensor field" or more commonly "tensors". These are objects which are pointwise in nature which live on a manifold. For example, functions and vector fields are examples of this. When I say pointwise in nature, I mean it doesn't behave like the derivative where you need local data to compute what the value at a point is.
A manifold is the underlying space these tensor fields are defined on. A manifold isn't a tensor field. For example, you can always add two tensor fields of the same type (e.g. you can always add two functions or two vector fields by pointwise addition), but you cannot (unless you manifold is $\mathbb{R}^n$) add two points on a manifold.