Does anyone know why Sera is the only companion you can ask to leave the Inquisition? I'm not sure if it has anything to do with your friendship level (as far as I know other companions have disapproved of my actions more than she has) but I'm near game completion and she is the only one who has the 'You don't fit the Inquisition' option available when I speak with her.
From my researches, there is no "official" answer to your question.
However, there are some character that can be told to leave :
- Cole (You can tell him to leave in one conversation)
- Sera (You can tell her to leave at anytime)
I don't know for sure about the other characters, but it say's here that if their approval drops low enough you can tell them to leave. (Unconfirmed)
I don't think you can get rid of Cassandra or Solas at any point.
There is no official reason. However, there are some lore-related reasons as to why Sera (and Cole at one point) can be asked to leave the inquisition.
The inquisition is an official organization recreated to fight the breach (and later corypheus). Sera is an outlaw, a Friend of Red Jenny, whose questionable legal status can damage how outsiders view the Inquisition.
Sera is also quite different in spirit from the other members of the Inquisition. Sera is impulsive and arrogant, witty and egotistical. She frequently pulls pranks on other people, goes into witty banter with both Solas and Vivienne and generally gives the impression of someone who doesn't care about rank or position.
Because of the above reasons, not everyone is equally happy with Sera. Vivienne and Cassandra in particular are not pleased with her being part of the inquisition. Most companions disapprove of the idea of the Inquisitor romancing Sera.
On a side note, if you choose to romance Sera, she can no longer be sent away. On a personal note, I think Sera is one of the best companions in Dragon Age: Inquisition. She's such a nice and welcome change from all the seriousness and gives the game some well-needed lightheartedness.