Why do we define addition of matrices only when they have the same size What happens if we define
$$
\begin{pmatrix}
        1 & 2 \\
 1 & 2 \\
        1 & 2 
        \end{pmatrix} +
\begin{pmatrix}
        1 & 2 & 3 \\
 1 & 2 & 3 \\
        1 & 2 & 3
        \end{pmatrix} =
\begin{pmatrix}
        2 & 4 & 3 \\
 2 & 4 & 3 \\
        2 & 4 & 3
        \end{pmatrix}
$$
I think computers do this to refresh a part of the screen. But why can't we do it by matrices by filling the remaining rows/columns with zero? Is it related to the definition of vector spaces?

Then you are just implying zeros in a third invisible column. If you really want to call that a 3x2 matrix (that there really is no last invisible column of zeros) then I would argue that your idea of subtraction is not a closed operation, which is not desirable. You would also have non-unique additive inverses, which is also not desirable.
There is nothing wrong with "doing it" but that is why it is not defined that way I suppose. It is rather like adding a 2d vector (a,b) to a 3d vector (c, d, e). You can say it is (a+c, b+d, e) if you want, but you didn't really add a 2d vector to a 3d vector, you added a 3d embedding of the 2d vector in the 3d space to a 3d vector. I think (a,b) in R² can be identified naturally with (a,b,0) in R³, but they are not the same -object.-
