Firstly, almost no-one actually creates their matrices as literals. Instead they are build dynamically an usually incrementally. Nevertheless, people still think back to these very basic examples when trying to understand things, so this probably should be standard.
Secondly, these decisions matter a lot for matrix vector multiplication. The 'C' way is very efficient for computing Ax which presumes vector x to be a Column vector. When you want to compute xA instead which presumes vector x to be a row vector, you want column-major ordering for efficiency. The same works for matrix-matrix multiplication AB where you want A to be row-major and B to be column-major.
To sum up. Because of our writing direction, and the natural way to write 2-d arrays, it makes sense to have row-major ordering. That immediately means it also makes sense to use column vectors, generally speaking. There are also exceptions if you happen to know if a matrix will often be at the 'left' or 'right' side of a multiplication.
Some more interesting notes that are less relevant:
In C the matrix you defined has an incompatible type with how most matrices are stored (most people in C store rows contiguously, then define a matrix as an array of pointers to rows).
int mat[3][3]; vs int *mat[3]
Another common ways to use matrices is element-wise, where you refer to x[i][j]. Mathematicians expect i to be the row and j to be the column. Which matches C. Whilst Fortran does it the other way around.