It seems to identify the wearer as a member of Sharalyan's 'Circle of Knowing', the renegade group of scholars and researchers from the Sharalyan homeland led by Louisoix. They were originally a separate group to Minfilia's organization, the later known as the Path of the Twelve which were regular Eorzeans who had become awoken to the Echo. After the Calamity though and Louisoix's disappearance, the remaining members in Eorzea sought Minfilia out realizing that those with the Echo would be the best ones to ally with against the revived Primal threat. Hence why Minfilia and the other Scions don't have tattoos - it's a legacy of the Archons' Sharalyan origins.