Here is an easy explanation for future readers (including that last confusing chapter)
Edna's ancestors (wealthy Duchaine Family) fought with Eli's family (McGreggors) and obviously won.
Many years later, Eli's family has all heirs named Hayden.
Eli's father Hayden wanted him to become King, and Cayden his brother was supposedly born to become a sacrificial tool to support Eli as King so that's why Cayden was named the heir of the McGreggor House despite being illegitimate
Obviously Eli never wanted to be King and now his younger brother is on the throne.
So NO Eli and Edna are NOT related but Edna has to hide her identity because she may be killed otherwise for being A Duchaine
This is why Edna's mother hid her true identity. Women in this time are mistreated( Edna's mother, the count's 10 year old daughter, Chris' mother, and Eli's mother). If they are lucky, they can find a good wealthy husband who will protect them. And even in the case of the husband being wealthy, he could just give you away. Case in point, Eli's mother.