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I only recall the youngest of the Coco brothers being stated to have a insufficiently sized mana pool or something, which suggests there's some way of measuring such a thing. And maybe something about novice Arcanists being tested for summoning Carbuncle. Nothing about any sort of procedure or anything for the non-WoL folks.
I always assumed it's something people who use magic can sense. The voidsent in that arc you mentioned with the coco brother made mention of sensing it. Given the nature of the arcanist guild it wouldn't be unreasonable to think they had standardized it in some way.
It could just be as simple as getting them to cast a basic spell, and seeing how well they do? Although that depends on exactly how magical ability works here - do spells need to be practiced, or will saying the right incantation give the same effect (modified by the individual's mana) whether it's a novice or a master?
Arcanistry seems to be off in a separate field to the black/red/white magic skills, but in the quest where Tataru tries to become an arcanist, Thubyrgeim mentions that Tataru's skill with numbers helped her grasp the concepts enough to summon Carbuncle even though she didn't have much magical aptitude.
I couldnt say if magical aptitude is a direct correlation to how much aether a person has but in many points in the story we see people talking about how much aether a person has. The scions also have their goggles which allow them to see aether directly and Y'shtola is able to "see" aether itself as well. I wouldn't be at all surprised if the thaumaturges and arcanists had a ritual or spell to cast that would give an indication of a persons power. We also know from the Alchemist quests that ink can react to a persons aether so it wouldn't be a stretch for them to have a potion/ink that would change depending on how strong a persons aether is.
My view has been that aether is the battery that drives all the spell craft and its only a persons training or temperament that dictates what they're good at; some people like their cute carbuncles, others like to incinerate things, and some just want to stone people.
Unless you're a pure blood Garlean then you have no magical aptitude. It's why they have magitech for their weapons. While probably favoring anyone who's not a pure blood and has a high aptitude for magic put into the military. I kind of feel that it's based on how hardy you are. For instance the reason machinist have those batteries on their hips is because the guild master wants anyone to be able to use their skills. The batteries draw in aether to be used with the more complex stuff. Also the thm guild receptionist doesn't seem to quite have a test, but she has turned away many who don't have the right stuff. That being the right attitude or not enough of an aether/mana pool to draw on depends on the person.
People who practice magic can sense it in others. If you're a Black Mage and you go talk to the THM guildmaster, he'll comment about how tainted your aura's become (at least in the English version), indicating practitioners of the arts have a Sight. I imagine the stronger the aether, the better the owner's Sight, with a bit of skill mixed in so they'll know what the heck they're seeing
This and its mentioned in the Blm quest-line that the Hearer from gridania can sense that you practice black magic. Thinking that you are a rogue practitioner of the arts and wanting to arrest you.