For starters, the distance of the curve to the line (I don't like the way it was termed, A line whose distance to a given curve, even though it amounts to the same thing) is measured in a distance perpendicular to the line.

Next, it is not true that the curve can never touch the asymptote. A sine wave, e.g., that is decreasing in amplitude by some amount, say 1/x, can be said to be asymptotic to the x-axis. It will "touch" the x-axis at 0, pi, 2pi,... It is said to be asymptotic to the x-axis because the envelope enclosing the sine wave is asymptotic to the x-axis, where the envelope would be equal to the pair of curves y=1/x and y=-1/x. See Ex. 1 in rav's link.