Understanding orthogonal regression In orthogonal regression, we are trying to minimize the distance from each data point $(x,y)$ to the fitted model.

My question is, how come that there is a distinction between independent and dependent variables in orthogonal regression?

In my naive understanding, the fit we are trying to achieve does not distinguish between $x,y$ - we want to find the line that minimizes the sum of distance to each $(x_i,y_i)$.
Nice question.

Short answer: the distinction between independent and dependent variables may not matter to your loss/error function, but it might matter for your modeling function.

Think about fitting a parabola with orthogonal regression.  Here, it's really important which variable is dependent and independent, because the function is not one-to-one.  So I'd say the distinction matters more for "modeling." 

In a simple case, such as fitting a line, then any distinction is probably more philosophical than practical.