This may be a bit underbaked bc I'm not a history expert, but I do believe that if you must view the Pointed Hats as analogous to any single real-world group, it would be the Christian church during the medieval period in Europe. They're somewhat sequestered from the general population (the Great Hall and ateliers vs. the Vatican and monasteries); have near-exclusive access to certain important information (the truth about magic vs. the language of the Bible and much of historical record) and the means of media production (Silverwood ink and casting seals vs. the labor and resources to copy manuscripts); are supported by taxes but do not seem to pay any (ch. 48); are relied on to perform individual charity, general social services, and paid services for feudal lords and monarchs (a central tension in ch. 39 and much of the Silver Eve arc); and of course, are punitive towards heresy and apostasy to the point of having a specialized judicial branch to root out sinful witches (the Knights Moralis vs. the Inquisitors, who are often depicted in similar colors).
WHA's setting is clearly inspired by the medieval period in its aesthetics and worldbuilding, but it doesn't include any organized religions (iirc) because the secular Pointed Hats roughly fulfill that role, including as moral arbiters for this society. Knowing that can also clarify the position of forbidden magic in the story as not just crime, but sin (as defined by Pointed Hat dogma). Despite sharing the name of "witch," Pointed Hats aren't at all comparable to witches as they were understood in the medieval period, while the Brimhats are functionally the witches in this analogy. They are marginalized thanks to their heretical beliefs and/or sinful practices (esp in cultivating knowledge preceding and/or contradicting the current world order) and offer a precarious alternative to the stifling conformist institution of the Pointed Hats.
Now, if you're reading this and going, "Hmm, okay, but WHA puts the Pointed Hats in a generally positive light and shows them as being far less violent, corrupt, and oppressive than the actual medieval Church, while also playing the Brimhats pretty straight as borderline demonic criminals rather than unfairly marginalized heretics or political targets," I think that's a good line of thought to follow! It's similar to the "does WHA actually say ACAB or do you just like Shirahama" debate but a bit more accurate to the Pointed Hats' actual role in this fictional society, imo (because as I've said in the past, I don't think WHA "says ACAB" by any metric, but I also think it's overly reductive to call the Knights "cops"). I also think this is a fun thought exercise if you, like Coco, fell in love with the sense of wonder and goodness around the Pointed Hats' magic, because we don't tend to look back in history and consider all the positive feelings the average person might've had towards the church, nor how radical and dangerous it would've actually felt to question them.