What would be wrong with defining matrix multiplication for matrices for the same size?

It has been defined.
Just use the same rule that you apply for multiplication of matrices.
A (order m by n)
B ( order n by k)
AB (order m by k)

Same order
A ( m by m)
B ( m by m)
AB ( m by m)

If you WANT to define matrix multiplication in the way that you suggest, then there would be nothing wrong with doing it like that, because you are doing the operation according to the definition.

However, the killer question is, what would it MEAN when you have done it?  Matrix multiplication is defined in the way that it is because the result is meaningful, and can be interpreted as the solution to a certain type of problem.

Performed in the way that you suggest, the result would be mathematical gibberish, and of no use whatever.  So I guess that is what would be "wrong" with it.

A very natural question; forgive me if I redirect you to a previous answer I made in the link below

To make a long story short: there is nothing wrong with that definition of matrix multiplication--- it is in use, although it is often given a different name to distinguish it from what is more commonly meant by "matrix product."  The reason the more common meaning of matrix product is what it is is that very often we think of matrices as functions, and the standard matrix product corresponds to function composition (the entrywise product does not).

To multiply two matrix, the number of columns of the matrix on the left has to equal the number of rows of the matrix on the right.

If A is n x m and B ix m x n, then AB = a x n

If you define A and B to have the same dimensions, i.e. A is m x n and B is m x n, then AB doesn't make sense since the number of columns of A = n and number of columns of B = m (they are not equal).