Suppose we want $ \begin{bmatrix} a & b \\ c & d \end{bmatrix} = \begin{bmatrix} p & q \\ r & s \end{bmatrix} \begin{bmatrix} a & b \\ c & d \end{bmatrix} $ to be true regardless of which matrix $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$ is, so that $\begin{bmatrix} p & q \\ r & s \end{bmatrix}$ is an identity matrix. Since it's true regardless of which matrix $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$ is, it must be true in particular if $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$ is $\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$, so we have $ \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} =\begin{bmatrix} p & q \\ r & s \end{bmatrix} \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}. $ This last equality clearly implies that $\begin{bmatrix} p & q \\ r & s \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$. Conclusion: if $\begin{bmatrix} p & q \\ r & s \end{bmatrix}$ is an identity matrix, then $\begin{bmatrix} p & q \\ r & s \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$. Therefore there is only one $2\times2$ identity matrix. And the same argument works for bigger matrices.