Matrix Multiplication = Matrix Concatenation?

What are we to make of the following matrix multiplication like this. At first sight the multiplication looks straightforward enough: $$\begin{bmatrix} 9 & 6 \\ 14 & 9 \end{bmatrix} \cdot \begin{bmatrix} 6 & 3 \\ 7 & 6 \end{bmatrix} = \begin{bmatrix} 96 & 63 \\ 147 & 96 \end{bmatrix}$$ However, we then see that the same result could have been achieved by concatenation of corresponding elements because: $$\begin{bmatrix} 9 & 6 \\ 14 & 9 \end{bmatrix} \cdot \begin{bmatrix} 6 & 3 \\ 7 & 6 \end{bmatrix} = \begin{bmatrix} 9|6 & 6|3 \\ 14|7 & 9|6 \end{bmatrix} = \begin{bmatrix} 96 & 63 \\ 147 & 96 \end{bmatrix}$$ where the | symbol denotes concatenation. Obviously this is not what normally happens when one 2 x 2 matrix is multiplied by another, so what special conditions need to apply in order for this equivalence of multiplication and concatenation to take place?

An explanation is provided in this source and is shown in Figures 1 and 2:

Figure 1

Figure 2

When the two matrices being multiplied are the same, the matrix can be called a "vampire" matrix. For example: $$\begin{bmatrix} 3 & 4 \\ 6 & 8 \end{bmatrix} \cdot \begin{bmatrix} 3 & 4 \\ 6 & 8 \end{bmatrix} =\begin{bmatrix} 33 & 44 \\ 66 & 88 \end{bmatrix}=11 \begin{bmatrix} 3 & 4 \\ 6 & 8 \end{bmatrix}$$ Note that multiplying one matrix by itself is equivalent to multiplying by a scalar (11) which is one of the eigenvalues of the matrix (the other is 0). 11 is also the trace of the matrix since it is equal to the sum of its eigenvalues. In November of 2019, I made a post titled Eigenvalues and Eigenvectors that explains these concepts.