The analogue of Jacobi polynomials in two-variables was introduced by Koornwinder, who described seven classes of orthogonal polynomials from Jacobi weights. Koornwinder also considered these seven classes as two-variables analogues of the classical orthogonal polynomials in one variable. The most commonly accepted definition for classical orthogonal polynomials in two variables was introduced by Krall and Sheffer. They generalized the characterization for classical orthogonal polynomials in two variables as eigenfunctions of a very special hypergeometric second-order differential operator, showing that up to an affine change of variables, there exist nine different sets of classical orthogonal polynomials. However, general tensor products of classical orthogonal polynomials in one variable and some of the Koornwinder classes are not classical, according to the Krall and Sheffer definition. Recently the authors extended the concept of classical orthogonal polynomials in two variables to a wider framework which includes the Krall and Sheffer definition, as well as the tensor products of classical orthogonal polynomials in one variable. This definition is the key to define the semiclassical orthogonal polynomials in two variables that the authors introduced in a recent work. Using their definition of classical and semiclassical orthogonal polynomials in two variables, in this paper, the authors study the classical and semiclassical character for the Koornwinder classes of orthogonal polynomials in two variables.They also provide the corresponding Pearson-type equation. In addition, using the Koornwinder construction, the authors provide some new examples of orthogonal polynomials in two variables.