"Choose side" in the ship right-panel is restricted to the Warzones or Conflicts zones, where you exit supercruise to find 2 opposing forces fighting each-other. You can then select a side.
Selecting a faction outside these zones have no function.
If you want to side with the faction however, you may start doing mission, trading or killing for them - you will then gain standing with them and see some progression through elite-ranking. Meaning you may get better offerings from them later, and a chance to participate in their factional missions / events on their side (later).