Most of the Shuttle electric charge was caused by electro-motive force (emf). A wire, when moved within a magnetic field, will experience flow of electrons. The Shuttle moves through and crosses the Earth's Magnetic field. If it trails a long wire, then you get a significant voltage and current.

A fixed space elevator would be static within the Earth's magnetic field, but fluctuations of the field during solar storms would cause significant currents in a conducting cable - just as with long electric transmission lines on Earth's surface - solar storms can set off voltage overloads, arcing and breaker trips.

The Earth's Magnetopause, on the daylight side, is at approx 70,000 km. Geostationary orbit is at ~42,000 km and at least some counterweight is required for the cable, so the end of the cable would be appreciably close to the daytime magnetopause. Every day-night cycle, the Earth's magnetic field is stretched and compressed from 70,000 km to many hundreds of km on the nighttime side. This cyclic compression and rebound would generate slow but perhaps strong currents in the cable (at equatorial lattitudes, the magnetic field lines are nearly parallel to the Earth's surface and would intersect the cable at a steep angle, but during compression and rebound, only a limited amount of magnetic flux would CROSS the cable).

If the end of the cable ever protrudes ABOVE the magnetopause, then that section of cable will be moving through the solar magnetic field at a very significant rate. Local current generation would be significant.

In theory, power can be tapped from tame electric impulses, but the unpredictable voltages and currents could be hard to harness safely.