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).
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).