Approaching the concept of a manifold from a physicists perspective there is one particular question that I have not been able to answer myself:
Is a manifold an intrinsically geometric object, or is the geometry purely specified by the additional structure of a metric?
For example, consider the unit sphere $S^{2}$. Without specifying a metric it is simple a set, along with a topology. Can one say that this is a geometric object because it is locally homeomorphic to $\mathbb{R}^{2}$? From an intuitive point of view, I can see that a geometric object such as a sphere can exist without the need to specify any sort of metric defined on it, but I'm not sure whether this intuition carries over to manifolds in general - is it even correct to say that $S^{2}$ is a unit sphere without introducing a metric on it?!
The question is, what you mean by $S^2$. If it is only a set, then you don't have much structure and its only property is the cardinality $c$.
If you can see the topology, then you know what are neighbourhoods, limits, homology and homotopy groups, Euler characteristic and some kind of "dimension" (although this is not trivial).
A manifold structure is even a richer structure (for example, the notion of dimension is much cleaner and more canonical). However, very often people think of smooth manifolds in which case you additionally get an idea of differentiability, tangent spaces and tangent vectors: this can still be done without a metric. All the field of differential topology deals with smooth structures and differential invariants that are, sometimes, defined via a metric, but are independent on the choice of the metric. By the way, on the manifold $S^2$, there is only one smooth structure up to isomorphism, which is not necessarily the case for higher-dimensional spheres.
But if you don't have a metric, you can not define distances and angles between tangent vectors in a canonical way. If this is "geometry", then no, you don't have it.