I think you have a good point! A vertical line also intersects the parabola at only one point. But I think the intent of this question is to find the slope of the line that is tangent to the parabola at (1,1). This is hinted in the question because it says "touches" instead of "intersects".
If you remember from geometry, a tangent to a circle is a line that just barely touches that circle. For circles, it's equivalent to saying that the line intersects the circle exactly once. But for other curves (like the parabola), you can imagine that it's much more complicated than that! In general, the idea behind a tangent line has nothing to do with how many times it intersects the curve. Rather, what matters is that the line "barely touches" the curve somewhere.
The method Professor Loh uses to find the slope of the line is actually one way to "define" what a tangent line is: You have to bring two points on the curve closer and closer together, and the line that connects them will "approach" a line that just "touches" the curve at a single point.
I'm glad you brought this up. This is often a confusing concept for some calculus students, and this all happens to lead to the extremely important calculus concept of the derivative!