From the Escaflowne Compendium:
Director Akane Kazuki says that Eries refuses all marriage arrangements by her father because she continues to love Allen. That is why she is still single and is not part of the royal lineage of succession.

From Animerica's Escaflowne Q & A:
Q: Why is Millerna's bethrothed the next in line to the Asturian throne, even though she's the youngest daughter?
A: King Aston had three princesses: Marlene, Eries, and finally Millerna. Normally, if there is no direct male heir, the betrothed of the oldest daughter will ascend to the Asturian throne. However, King Aston married off his oldest daughter Marlene to another country's ruler. Then why is Millerna being positioned as the wife of the next king, and not Eries? Eries has her heart set on someone she can't marry....

This succession deal is beyond me. King Aston strikes me as a chauvinist because he won't let his own daughters rule, they have to marry the next king. And i used to wonder if perhaps Eries was to be sent off to some nunnery. ^_^ But from the director's spoiler i surmise that she simply will not let her father maneuver her into a political marriage to some nobleman in a neighboring realm, like he did to Marlene. To me this implies that Eries has greater strength of character and will than Marlene. In any case it's unfortunate that Eries isn't in line for the throne of Asturia, because i think she would make a great queen. There's something very regal, dignified, Elizabethan/Victorian about her.

~Main~ ~Consort~