What is the cartesian product of cartesian products?

By definition $(A\times B)\times(C\times D)$ is the set of all ordered pairs of ordered pairs of the form

$$\big\langle\langle a,b\rangle,\langle c,d\rangle\big\rangle\;,$$

where $a\in A,b\in B,c\in C$, and $d\in D$. You’ve described the set of $2\times 2$ matrices of the form

$$\begin{bmatrix}a&b\\c&d\end{bmatrix}\;,$$

where $a\in A,b\in B,c\in C$, and $d\in D$. There is a natural bijection between these two sets given by

$$\big\langle\langle a,b\rangle,\langle c,d\rangle\big\rangle\mapsto\begin{bmatrix}a&b\\c&d\end{bmatrix}\;,$$

but the two sets are not equal, simply because a $2\times 2$ matrix is not an ordered pair of ordered pairs: the set of matrices is not the same thing as the Cartesian product.

Cartesian product of cartesian products is a set of tuples of tuples. $$(A \times B) \times (C \times D) = \left\{((a,b),(c,d))|a\in A, b\in B, c\in C, d\in D\right\}$$

$(A \times B)\times (C \times D))=\{(u,v): u \in A \times B, v \in C \times D\}=\{((a,b),(c,d)):a \in A, b \in B, c \in C, d \in D\}$.