Kronecker Delta as a Tensor

Question

Let $\delta^i_j$ be the Kronecker delta function, i.e. $1$ if $i=j$ and $0$ otherwise. Then, it is easy to verify that this value is a rank 2 mixed tensor of one covariant index and one contravariant index. But, what does this mean intuitively? How does it make sense to interpret $\delta^i_j$ under a change of basis? The function is the identity matrix which is invariant under any change of basis.