Thinking of a configuration of 3 lines as a cubic plane curve, we can represent such a configuration by a degree 3 homogeneous polynomial which factors completely into linear factors, modulo scalars. This naturally embeds such configurations into the projective space of cubic forms $\mathbb{P} H^0(\mathcal{O_{\mathbb P^2}}(3))$, a 9-dimensional projective space.
The general configuration of lines, consisting of three lines meeting at three distinct points in the (finite) plane forms a 6-dimensional (quasiprojective) subvariety of this projective space, as it is the image of the finite map $\mathbb{P}^{2*}\times \mathbb{P}^{2*}\times \mathbb{P}^{2*} \to \mathbb{P}^9$ given by unioning three lines in the plane.
When three lines all meet at the same point, the configuration is determined by selecting the point of intersection and the slopes of the 3 lines. This is therefore a 5-dimensional locus.
If two of the lines become parallel, the configuration is essentially determined by choosing the slope of the parallel lines (equivalently, the point they meet along the line at infinity in projective space) along with say the $y$-intercepts of the parallel lines and the position of the third, nonparallel line. This gives a 5-dimensional space as well.
Other degeneracy loci can be analyzed similarly, essentially by determining the number of degrees of freedom in a configuration.
Schubert calculus and intersection theory don't seem particularly relevant, as these degeneracy loci aren't Schubert cycles and aren't naturally described as intersections.
