[defrost]

Birkeland currents flow following magnetic field lines.

The Earth's magnetosphere is shaped by solar winds, rounded on the sunside extending 6-10 x Earth radius with a longer tail opposite the sun that extends up to 1,000 x Earth radius.

The Earth's radius is very small compared to the 1 AU Earth distance from Sun and smaller again compared to the distances between planets.

So, rather than "filling the entire universe" the currents you've linked flow through a magnet field that rapidly dwindles with the inverse square of distance and neither the electric currents nor the magnetic field reach to another planet (nor vice versa).

[pyinstallwoes]

Plasma. It's everywhere.