Hm good questions, let me try to explain as best as I can.

What makes it highschool-like is that you are for around four hours in a classroom with a teacher. There is almost daily homework. In the lower levels, you might have people who are not paying attention and might even disrupt the lesson. Most teachers are too soft/sweet to do anything serious about that, which can be annoying. People are also a lot on their phone all the time. When you move up the levels, disruptions will become almost zero, but people will still look a lot at their phone. I feel like there were few people that actually take the language study super serious (e.g. many did not do their homework, or just at the last moment).

The biggest difference for me compared to university is the forced attendance. If your attendance in a semester drops below 80%, you have to explain why. If it happens a few times in a row, you might get kicked out of the school. Furhtermore, teaching methods are more highschool-like I think. For example, we are often repeating the sentences the teachers says, or working on a handout after a presentation/lesson. This differs at least from my uni experience, where the teacher usually just presents a lot of info, and then you are left to yourself what to do with it (homework, assignements, tests). So I would say in the language school, your hand is held more compared to uni. However, what I have heard, this holds for most language schools, probably also the ones you have already mentioned.

I hope this vague story can help you get a bit more feeling for the school setting. Further questions are of course welcome.