Thom - You Asked - and there's a bit of history to behind this one. Also a funny sort of co-incidence, as I am, this moment, actually tinkering on the side with a storyline from here…grin.
Before refuge on Athera, since the order's founding, initiates to what is now the Koriathain give up their past and their families when they come to the order. Originally, they actually gave up their former names, for new ones given them by the Order. These became the names they went by, with the order considered their only tie - their only loyalty, in relationship. Therefore, they only had one name for their individuality. If this were so, today, and they had to sign legally, the signature would be: Prime Matriarch, Morriel Koriathain. Or First Senior Lirenda Koriathain. Initiate Elaira Koriathain. Legal "family" status was acknowledge as Koriathain.
When the Order arrived on Athera, this practice of completely giving up the birth name was forced out of practice by the compact. Paravian Law held that the soul named the child, and that facet of life should not be dropped. The birth name must therefore remain an inviolate part of identity. So the practice on Athera had to be amended to compromise - the surname was dropped, and only the given name from birth retained in use within the order.
There is an induction ceremony held, before the formal oathswearing, wherein the surname and the past ties to family are "ritually forgotten" - and no surname is ever pronounced after that. It is effectively renounced and eradicated.