In the projective plane P2(K), where K is field, is the line passing through two distinct points not unique?

Question

So, we form the coefficient matrix, which is a and the rank of this matrix is two which leaves one free variable, which means the coefficients of the line are not unique. But this is in direct contradiction with the axioms laid out for projective planes.

Answer 1

… the coefficients of the line are not unique.

This is true: the coefficients of the line are not unique, but the coefficints are merely one representation of the line. You may scale all coefficients by a non-zero scalar factor and obtain a different representation of the same line. That's what homogeneous coordinates are about. Formally speaking, the line is an equivalence class of all these non-zero multiples, and that equivalence class is uniquely determined by two distinct points, even though you are free to pick an arbitrary non-zero representative to represent the class.

Note that the same holds for points, too: points represented using homogeneous coordinates are identical if their coordinate vectors only differ by a scalar factor. So when you think about the line joining two points, you have to ensure that these two points are actually geometrically different. Otherwise the coefficient matrix would have rank one.