Affine of Inverse Sboxes
To be precise, different affine transformations following the Galois field inversion should give the same Walsh spectrum up to a $\pm$ sign, in terms of how many times each value occurs. Since an affine transformation is a linear transformation plus a constant vector addition, this is not surprising.

Don't forget linear cryptanalysis measures distance to the unbiased case (prob. 1/2) so signs of Hadamard coefficients switching between positive and negative is allowed. This is how come we can ignore the sum of the non targeted key bits modulo 2 during Linear cryptanalysis, since all they would do is switch the sign of the relevant coefficient.

Let 
$$L_{a,b} := \sum_{x \in V_n} (-1)^{a \cdot x \oplus b \cdot S(x)}$$

where $V_n$ is the n dimensional binary vector space. Let $A x+c$ be an affine map where the linear part $x\mapsto Ax$ is full rank and thus invertible. It is then a simple matter of algebra to prove the result:
$$L_{a,b}' := \sum_{x \in V_n} (-1)^{a \cdot x 
\oplus b \cdot (A \cdot S(x)\oplus c)}$$
by a change of basis.

Even better, in "The Design of Rijndael" by Daemen and Rijmen, (See  here, Appendix A.1 onwards, provided for personal research use only) there is a coordinate free approach using trace functions on the finite field to show this.

Similar comments apply for the correlation spectrum.