According to the Crystal Sphere Techniques, even the flexible magic of a mage is still limited to a single sphere of influence. Archmages are ones who are capable of tapping into multiple spheres to cast their magic. It is a glorified term for a capable mage who can use multiple elemental, reactional, or even more esoteric magic from the spheres of magic.
Like the inspiration from the Victorian Age, no one has a real clue how magic works. They have ideas and theories, usually shouted from the rooftops so everyone can hear them. The Crystal Spheres is just one of many systems that includes clan magic, tiags, and everything else. It also happens to be popular and being known as an “archmage” is not unlike being known as a CEO or influencer or rich person, just the label is sometimes enough to open a door and give opportunities.
It also means that archmages are subject to additional scrutiny as others try to tear them down.
Right now, there are only two characters who (will) have a label of being an archmage, both are in unwritten books. If all goes well, one will be in Pack Daughter and the other will be in Her Silken Touch, my attempt at a romance novel that doesn't turn into polygamous sex scenes.