Tested for a few days, on 2 characters.
Setting up a respawn point for your companion does not work. Most common case linked above (no reaction from a companion).
On a few occasions I had a dialog box saying something like "set to '' ". On one occasion it said something like "no such base found" .
All of the typed bases exist in our fraction.
I have tried confirming an input with both "ok" button and enter.
I have also tried setting a respawn point while standing at the target base, any other base, and in a wilderness. Nothing works.
Setting a tent as a respawn point just takes me back to the companion's conversation main menu.
If a companion is killed, he respawns at the first base the fraction had bought (script build-in emergency respawn attempt if I remember it right ).