Sorry if this is a stupid question (I'm a physicist, not a mathematician), but can you explain why we say that a geometry is symplectic rather than that we can just define both a metric and a cometric? As a concrete example, if I consider a 2D Euclidean geometry, I can define both the ordinary inner product, as well as a symplectic inner product, and each might be separately useful in different ways on the same geometry. What is different from this viewpoint and a geometry itself "being" symplectic?