Do you think it's fair to say that some aspects of magic with ASoIaF have been, or shown signs of being, systematized? I think skinchanging is the best example: we have a bunch of POV chapters from skinchangers, including Varamyr's prologue which is brimming with information on the subject.
I wouldn’t say so, no, sorry. Systematized magic is, well, you put 250 fluid ounces of water and 5 ounces of African sea salt in a silver beaker, add 10 drops of essence of wormwood with your right hand and then another 10 drops with your left hand, stir 10 times clockwise, heat at 200° for 15 minutes, etc, etc. If becoming a skinchanger required that a child from a certain race needed to be exposed to a certain animal by a certain age for a certain amount of time, with certain potions to drink to put him into the right state of mind for bonding, with specific magic words he needs to say to transfer into an animal’s head, with an established school that potential skinchangers were required to attend and read books about what animal should be assigned to which kind of person and learn the magic words and how to brew the skinchanging potions – and most importantly, if all these rules were not followed precisely, then skinchanging wouldn’t work – then, yes, it would be systemized.
Instead what we learn from Varamyr’s chapter is that a young boy from a wildling family started skinchanging with one of the family dogs by pure magical instinct, and after his abilities were discovered (after he used the dog to kill his little brother, and screamed when the dog was put down), he was abandoned by his parents and sent to a local warg, because he wasn’t fit for normal human society. The warg introduced him to other local wildling skinchangers, and endeavored to teach him what he knew about skinchanging by experience and lore: (a) what animals were easiest to bond with and which were the hardest, (b) that the animal you bond with affects your personality and therefore he didn’t think some animals were good to skinchange into, (c) the “abominations” that skinchangers hold are forbidden to do, and (d) what happens to skinchangers after they die.
But notably this is all custom and oral tradition, not rules. If a skinchanger does one of the abominations – say, stays inside their animal while it’s having sex – it’s not that skinchanging stops working, it’s that other skinchangers will think you’re a horrible person if they find out, and maybe the gods will think that too. Maybe it’ll affect your mind in some way, maybe you’ll change in ways you won’t like. (Maybe you won’t even realize how much you’re changing.) But the important word there is maybe. For all Haggon believed that skinchanging birds was the worst, that those who’d tried birds ended up staring at the sky all day only wanting to fly, we’ve encountered several bird skinchangers who appear to be quite centered and not “moony” at all.
There aren’t systemized rules for becoming a skinchanger – you don’t have to drink a potion or say magic words, you form a connection to an animal, feel their feelings (and they feel yours), and one day you just fall asleep and dream you’re the animal. And as you grow stronger, you can reach out and move into an animal’s mind deliberately, and later any animal, not just the one you originally bonded to. But how do you do it? You just do it. It just happens. It’s by acts of will and acts of instinct, not by a system. Some things make it easier – if an animal has been bonded before, if it’s a domesticated animal, if you have someone else guiding you and explaining things. But it could be someone would follow everything exactly and never be able to touch an animal’s mind, because they weren’t right for it, the magic wasn’t right for it, who knows. (And I wouldn’t doubt if some maester tried it and got nothing, and came away believing that skinchanging was never real in the first place.) It’s not a system, it’s not a science, it just is.
Hmm… I think the best comparison is to wizardry and witchcraft, per Terry Pratchett’s Equal Rites. Wizardry is very precise, very systemized, words and exact potions and stars and books and geometry, taught in a regimented hierarchal school with rules for entry. Witchcraft is instinctual, intuitive, emotive, recipes that are just a pinch of this and a drop of that (or just a colored water placebo), drawing on psychology and just knowing how people work, taught individually to an apprentice or just by picking things up all by yourself. (And notably, Granny Weatherwax’s “borrowing” is straight up skinchanging under a different name.) Skinchanging, all of the magic of ASOIAF, is far more like the latter than the former.
I hope that helps!