Below, what we learned about the story of the job:
1) Where it came from: The Whalaqee is a tribe of hyur from the New World and they have a tradition of casting magicks that can mimic the abilities of creatures. Martyn named it Blue Magic, but originally it might have a different name. He did so because of the Whalaqee's reverence of the color blue and the meaning it represents for them. (Also Martyn is biased in having a love for the color blue as well)
2) Tuxedo mask:The Blue Mage aesthetic presented, with the dapper blue suit and mask is something that Royse imposes Martyn to wear (and consequently you, later on), to create a character bound to display magick showdowns on an arena. We also see a couple of images of Whalaqee spellcasters wearing more traditional outfits, in blue.
3) What it feeds on:Blue magic draws aether from the surroundings, like conjury. Though it's not explored about possible harmful effects. I speculate it could be a future plot point for the job quests, when you might harness advanced abilities that might draw more dangerous amounts of aether.
4) How it works: Blue magic attempts to mimic the best it can from the monster abilities. It usually uses aether to shape the material components of the spell, like the fish hurled by "Flying Sardine". The description of the spells are a nice insight on how the job functions mechanically.
5) About the weapon: It's briefly mentioned that the cane is a magic focus, but not any further insight if it necessary to be crafted using something specific, like THM or CNJ arms.
6) About the soulstone: The BLU soulstone, according to the tooltip, functions like MCH's. An empty vessel to record knowledge.
7) Future: Martyn contemplates the possibility of opening a Blue Mage guild as soon as financially possible, to spread the knowledge, so we might see it as an actual organization in the future.
8) How one becomes: How a regular adventurer might become a Blue Mage? The most straightforward route is simply coming across Martyn and purchasing his wares (before he got prohibited, of course), or in the future by enrolling in a possible Blue Mage guild.
Unlikely, but there might be the chance of learning in the source (New World) too.
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I might add more as I find new information.