In probability theory covariance matrix denote how each variable relates to other in a pairwise manner. So 1 would mean they are identical and 0 would mean they are independent and are not related. Is this concept similar to Jacobian matrix where such relation between multiple variables are denoted by partial derivative?
I'd rather say that the covariance matrix is similar to the Hessian matrix .. Furthermore, the Hessian tends to be proportional to the precision matrix (which is the inverse of the covarinace matrix). The intuition behind is that the higher the curvature in a given direction is, the lower the auto-covariance (the variance) in that same direction will be.
Consider the direction of $x$ axis in the figure below. At a fixed $y$ ($y = 15$ in the example) the sample of points $x_i$ drawn from the blue plot has a higher variance than the red sample. However, in the same direction $x$, the red plot has a higher curvature $.6$ compared to the blue one $.2$.