I formally understand what a riemannian metric $g$ on a manifold $M$ is. It's basically a section of the vectorbundle $T^*M\otimes T^*M \to M$ (which in the end corresponds to mapping 2 vectors from a tangent space at a base point to a real number after writing it out), that's symmetric, and positive definite. Bilinearity follows from this, therefor the riemann metric $g$ is basically assigning an inner product to each tangent space.
However, I do not have an intuition of the total. Simply having an inner product on each tangent space is understandable, but not the total structure has to be smooth, aka we know that sections of vectorbundles have to be smooth. I don't have a geometric intuition for this. For example for a vector field we have that at each point you have a different tangent vector, but the smoothness in this case basically says that you don't have "strange jumps or whatever" for the vectorfield (basically the flow is smooth). How about the riemannian metric, is there a similar analogy, and could it be explained?