Though the graph of the Dirichlet function is non-drawable, I think if we have to draw it in some informal way then it will be two complete lines (instead of isolated points).
Here's my reasoning: there cannot be any gaps on each of the two lines, because if there are such gaps, let the $x$-coordinates of the ends of the gap be $a$ and $b$, then the existence of the gap shows that there's no rational (or irrational) number in the interval $(a, b)$, which is absurd. Hence there are no gaps, and thus the lines should be complete.
But if the graph consists of exactly two lines, then it seems that the function is multi-valued. I know that all my discussion so far is concerned with informal concepts that deny an explicit definition. But my question is am I correct (in some sense) and how do I understand this "dilemma"?
What this boils down to is that you (and, given certain metaphysical assumptions, the universe) can't tell by looking at a line if there is a single point removed. It follows that you can't tell if a finite number of points have been removed, as so by taking limits you can't tell if countably many points have been removed. But just because you can't tell the difference between two lines doesn't mean they are the same. The graph of the dirchlet function is not two horizontal lines, and the function is not multivalued.
Your faulty logic lies in the assumption that something necessarily is what is looks like the second paragraph is about something that looks like a line. The third paragraph is about something that is a line.