What is the problem in "closest pair problem" if all points share the same x-coordinate The closest pair of points problem deals with the task to find a pair of points with the global minimum distance. There is a problem, when all points share the same x-coordinate, or at least a large number of points.

I just don't know why, I heard the running time becomes $n^2$

What about it? What is the problem, since the proof for the algorithm shows, that at max. 8 points can reside in the $2 \delta \times \delta$ area.
Points with the same x-coordinate do not cause any substantial problem.  However, if you implement the divide-and-conquer algorithm carelessly, they may cause a problem.  One way to deal with them is by using symbolic perturbation.