You're sort of correct, except keep in mind that its the same infinity in both directions, so we have four lines coming together at a point, and two lines spanning the plane. Kind of (exactly) like straps of a backpack wrapped around your body.
What you're discussing is different, though! You're talking about metric spheres, (or pseudometric spheres, in this case) which are the set of points a given distance (number of semitones) away from a given point. It's not a "sphere" geometrically or topologically, it just has that property of containing all the points a given distance from the "center."
You get two lines because each string is identical to the last, but shifted by a fifth (or fourth, depending on your direction). The "current" string has two points (count off your interval up and down. Each other string has those same two points shifted by five frets (except for that pesky g-b), so you get two points there.
Your space is discrete (each point is "separate" from the others, rather than connected by a continuous array of points) so no subset of it at all is homeomorphic to a Euclidean circle. You do, however, have metric circles.
On last point! The reason you have a pseudometric, not a metric, is becauase metrics require that "no two points lie at the same point." For any point, there are infinitely many other points (one on each string) that are "0 semitones away." In a metric space, only a point and itself may be 0 distance apart. This means that circles of radius 0 in metric spaces are just a point. In your space, the circle of radius 0 is a single line, since the two parallel lines come together.
