Example 4 Find all pairs of positive integers $(m, n)$, such that it is possible to color each cell of an $m \times n$ grid in white or black, and satisfy: for each cell $A$, the number of cells that share at least one vertex with $A$ and are of the same color as $A$ (including $A$ itself) is even. ${ }^{[3]}$
(2005, Silk Road Mathematical Competition)
A coloring that satisfies the conditions is called "good", and two cells that have at least one common point are called "adjacent", each cell is adjacent to itself.

If $m n$ is even, assuming there are an even number of rows, the 1st and 2nd rows can be colored white, the 3rd and 4th rows can be colored black, the 5th and 6th rows can be colored white, and so on, alternating the coloring. It is easy to see that such a coloring is good.

If $m n$ is odd, assume there exists a good coloring, then either the number of white cells is odd, or the number of black cells is odd. Without loss of generality, assume the number of white cells is odd.

Next, consider the set $W$ consisting of all ordered pairs of two non-adjacent white cells.

Since each cell is adjacent to itself, if $(A, B) \in W (A \neq B)$, then $(B, A) \in W$.
Thus, the number of elements in set $W$ is even.
Since the number of white cells is odd, for each white cell, there are an odd number of white cells that are not adjacent to it (because the number of adjacent cells is even), thus, the number of elements in set $W$ is odd, which is a contradiction.

In summary, the pairs of positive integers $(m, n)$ that satisfy the conditions should have $m n$ as even.