If I write a vector v = [1, 3, 2], what I am actually saying is that v is equal to 1 * e₁ + 3 * e₂ + 2 * e₃ for three vectors I have previously decided on ahead of time that form an orthonormal basis of the corresponding vector space.

If I write a matrix, say, this:

  [[1  2]
   [3  4]
   [5  6]]

> The natural motivation of matrices is as representing systems of equations.

That is very useful for only very few things about matrices, primarily Gaussian elimination and related topics. Matrix multiplication--which is what the original poster was talking about, after all--is something that doesn't make sense if you're only looking at it as a system of equations; you have to understand a matrix as a linear transformation to have it make sense, and that generally means you have to start talking about vector spaces.

Doesn't make sense is too strong though.

If you have a system Ax=y and a system By=z there exists a system (BA)x=z

This system BA is naturally seen as the composition of both systems of equations

And the multiplication rule expresses the way to construct the new systems' coefficients over x constrained by z.

The C_i equation has coefficients which are the evaluations of the B_i equation over the A_k-th coefficients

C_ik = B_ij A_jk

concretely

        A11 x1 + A12 x2 = y1
        A21 x1 + A22 x2 = y2

        and

        B11 y1 + B12 y2 = z1
        B21 y1 + B22 y2 = z2

        then

        B11 (A11 x1 + A12 x2) + B12 (A21 x1 + A22 x2) = z1
        B21 (A11 x1 + A12 x2) + B22 (A21 x1 + A22 x2) = z2

        rearrange and collect terms

        (B11 A11 + B12 A21) x1 + (B11 A12 + B12 A22) x2 = z1
        (B21 A11 + B22 A21) x1 + (B21 A12 + B22 A22) x2 = z2