Reason for thinking of vector as "row" and "column" vectors in linear algebra

First, there is usually not made any formal difference between the two points of view: there is only one notation $K^n$ for the space of $n$-tuples of scalars, not separate ones for those tuples written vertically or horizontally (or maybe in other forms still).

However, once one uses matrices to represent linear maps, one has to make the choice whether to organise them so that the coordinates (at arrival) of the images of the vectors of the source basis used are given by its columns or by its rows. The most common convention is to have the images correspond to the columns of the matrix, but some authors (in my experience mostly those who do not rely heavily on matrix calculations in the first place) use the convention to use the rows of the matrix. Supposing columns are used, the effect of a linear map on the coordinates of the vectors it operates on is given by left-multpication by the matrix of the linear map, and that operation requires the coordinates of the vector acted upon to be written as a column vector in the matrix product. So of $A$ is the matrix, with respect to chosen bases, of a linear map $f:V\to W$, and $v\in V$ has coordinates $(v_1,\ldots,v_n)$ on the chosen basis of $V$, then the coordinates of $f(v)$ in the chosen basis of$~W$ are the components of the product $$ A \cdot \begin{pmatrix}v_1\\v_2\\\vdots\\v_n\end{pmatrix}. $$ If we had written the coordinates as a row vector, then the only way to multiply by a matrix is to do so on the right (and by the transpose of $A$ if it is not square), but that simply does not give the right coordinates of $f(v)$. So writing column vectors is really just an easy way to remember what convention was used to represent linear maps by matrices.

Although at almost every point where a choice is made there are some authors that choose the opposite convention (and this even applies to the question whether the row or column index in a matrix is written first), I think that fortunately everybody agrees about the rule to multiply matrices (taking rows on the left multiplied by columns on the right). The above is based on the assumption that this convention is universal. Changing that convention would be a most certain way to be misunderstood by everyone. Moreover if one would define matrix mutliplication in a way consistent which what you write in the question, then as user68061 observed you loose associativity of multiplication, which means that matrix multiplication no longer corresponds to (linear) function composition, and it in fact become a completely pointless operation.

If you just have a tuple of numbers then, as you said, there is no difference between column and row. However, if you want to have nice matrix algebra, you have to distinguish between two.

The reason nice matrix algebra exists is the fact that, any $m \times n$ matrix can be understood as a linear map from $K^n$ to $K^m$. In this formalism, a column vector is a map from $K$ to $K^n$ i.e., a vector; but a row vector is a map from $K^n$ to $K$ i.e., a functional.

If you define the multiplication as you did, you will lose associativity, and that is bad.