That link doesn't contradict the person you're replying to. Actual orthogonality still has a probability of zero, just as the equator of a sphere has zero surface area, because it's a one-dimensional line (even if it is in some sense "bigger" than the Arctic circle).
If you're picking a random point on the (idealized) Earth, the probability of it being exactly on the equator is zero, unless you're willing to add some tolerance for "close enough" in order to give the line some width. Whether that tolerance is +/- one degree of arc, or one mile, or one inch, or one angstrom, you're technically including vectors that aren't perfectly orthogonal to the pole as "successes". That idea does generalize into higher dimensions; the only part that doesn't is the shape of the rest of the sphere (the spinning-top image is actually quite handy).
If you're picking a random point on the (idealized) Earth, the probability of it being exactly on the equator is zero, unless you're willing to add some tolerance for "close enough" in order to give the line some width. Whether that tolerance is +/- one degree of arc, or one mile, or one inch, or one angstrom, you're technically including vectors that aren't perfectly orthogonal to the pole as "successes". That idea does generalize into higher dimensions; the only part that doesn't is the shape of the rest of the sphere (the spinning-top image is actually quite handy).