Example 7 Fill in each small square of a $9 \times 9$ grid with a number, with no more than four different numbers in each row and each column. What is the maximum number of different numbers that can be in this grid? ${ }^{[4]}$
If there are 29 different numbers in this grid, by the pigeonhole principle, there must be a row with four different numbers (let's assume it is the first row).

The remaining 25 numbers are in rows $2 \sim 9$. Similarly, by the pigeonhole principle, there must be a row with four different numbers (let's assume it is the second row).
Next, observe the numbers in each column of the grid.
Each column already has two different numbers at the top. Therefore, each column below the second row can have at most two different numbers. This way, the maximum number of different numbers is $8 + 9 \times 2 = 26$, which is a contradiction.
Therefore, the number of different numbers is less than 29.
Table 1 constructs a $9 \times 9$ grid containing 28 different numbers, with each row and column having exactly four different numbers.