Malakhim are born from the world's concentrated energy; consequently, the concept of biological family does not apply to them. However, when Edna was brought into the world and Eizen's arms, he felt a special connection and swore that he would protect her no matter what. He raised and loved her like family, and he was devastated upon realizing that his blessing had caused her injuries and misfortune over the centuries.
Tales of the Rays, a mobile game whose premise involves projecting facets of foreign worlds, features an event where Eizen and Edna meet again. Their reunion takes on a melancholic twist when Eizen walks out on Edna, refusing to allow her company out of paranoia that he will bring her to harm. His sister, meanwhile, claims that he hasn't changed at all.
The crushing guilt of what he'd unknowingly done to his sister left a deep scar within Eizen, who partially refuses to return to the sacred mountain on which she dwells for her safety. While the swashbuckling lifestyle of a pirate cushions the blow of distance, Eizen's self-imposed exile still wears on him with the knowledge that he is hurting his sister's heart. His obsession with keeping Edna safe is the height of his reason.
Eizen suggests that Edna is the more emotionally mature sibling, and this is true. In Rays, his insistence on keeping distance causes the both of them a tremendous amount of pain; however, Edna deigns not to say anything, because throwing a tantrum is "what children do" and she sees how hurt Eizen is by his own actions. If it means keeping Edna out of his domain's reach, Eizen is more than willing to crush his emotional needs under the dominating heel of reason.
Throughout Berseria, Eizen paints the portrait of a sickly girl whenever he mentions Edna. To drive this image home, he named her after a dainty flower. As Edna is shown to be headstrong and resilient in Zestiria, if indeed a little dainty, the apparent disparity begs to differ on the prescription of Eizen's rose-tinted glasses and whether he realizes that he's probably responsible for her being prone to injury; if so, it only adds to the reasons as to why he won't go near her.
In fact, Eizen is so protective of Edna that he tells anyone who asks about her, "If you touch my sister, I will kill you"—and all in one breath, to boot.