@hdmi1 most likely that is a school blazer yes. But I don't think she is an isekai protag who invented them. The world does seem touched by isekai protags tho (english phrases like "lets rock" etc.), so I bet someone saw lack of schools and managed to weasel in with the nobility to get funding to start an educational facility inspired by their own past. But as another said, I doubt it is education for everyone, most likely only pertains to upper classes, and if higher enough, their vassals.
I bet they had private tutors too. Schools were as much for the connections etc., as they were for actually educating people. Though to some poorer rich classes the education probably were a bigger part of it too, due to not affording hiring teachers at home in addition to the school.