I believe so, though the ellipse is a “coincidence” of sorts, and not related to the more familiar solution of the two-body problem.
Your fall to the other end of the shaft will trace one half of a very long but very narrow ellipse, if you begin falling with some small non-zero lateral velocity (one not large enough to slam you into the side of the shaft though). Otherwise of course, if you start with zero lateral velocity, your path will be a straight line.
I’m making the same assumption the earlier web page author does: that the earth has constant density, and that therefore your acceleration vector is proportional to your position vector. (As Chronos points out, this might not be such a good assumption for the real Earth.) In this scenario, your acceleration toward the earth’s axis is proportional to your distance from that axis, much like it is with regard to the earth’s center, and with the same constant of proportionality. Therefore your lateral motion is also sinusoidal, with the same period as your motion along the axis. Therefore, you move along an elliptical path — though one whose center (and not a focus) is at the earth’s center.
The “ordinary” formulae you’re thinking of assume point masses of a fixed magnitude — or spherically symmetric masses, which can be treated as point-like as long as you’re outside their boundaries. In the OP’s problem however, the amount of mass that influences your trajectory is constantly changing as your position changes.