The key to making sense of this apparent exception is to recognize that in this geometric world, a line that goes off in one direction will reemerge from the opposite direction as it passes through a point on the line at infinity. So the "two" points you mention are not two distinct points, but the same point—just reached by going in opposite directions. If you could see far enough in this world, beyond and through that ideal point at infinity, you would actually see the back of your own head. So a line is still infinite in length, but it "loops back" onto itself.

When you stand in the middle of a road that goes on for ever, you see a "vanishing" point on the horizon (where the road seems to disappear). But if you turn around and look down the road in the opposite direction, the road emerges from a point on the horizon—which is the very same point into which it vanished.