In both the manga and anime, Flamme insists that Frieren call her "Master" instead of by name.
I don't recall anyone discussing that matter here, so here is my theory. I apologize if this has been discussed before or if it's obvious to you.
Flamme seems quite knowledgeable about the psychology of other species, such as demons and elves.
Specifically, she clearly understood the struggles Frieren would face with the deaths of her loved ones.
I think she knew that if Frieren grew too attached to her, Frieren's grief over her death could lead to total isolation for the rest of her life.
So, she planned to maintain some distance from her apprentice, with moderate success, while planting the seeds for a future world in which Frieren could experience grief in a less traumatic way.
In my opinion, it explains the "Master" thing and is a nice detail to their relationship.