Why the hell would it be a nickname?

[She can't imagine anyone going around calling themselves Mordred if it wasn't their actual name, and the confused look on her face says as much.]

It's my name. Saber's more like my nickname. I guess? It made sense at the time since I wasn't trying to throw my name out there to everybody.

[She says this like it all makes sense and isn't canon specific garbage that only applies to her here.]