For matrices $A$ and $B$, I thought $\frac{\partial A}{\partial B}$ is a matrix $C$ where $C_{ij} = \frac{\partial A_{ij}}{\partial B_{ij}}$.

However, when I use this matrix calculus website, it says

$$\frac{\partial{A}}{\partial{A}} = \mathbb{I}\otimes\mathbb{I}$$

Why is it a tensor? How to select the elements from this tensor to get the kind of derivative $C$ that I thought about?