For what it's worth, I'm a father of 3 and our oldest is in school 2 years now and could easily be described the same as yours. However, he is in a "good" school and we still see all kinds of new behaviours being brought home. Personally, I see it as being natural for his behaviour to change while experiencing all these new social dynamics. I see him having to go to school, put on a new personality that fits into the existing social dynamics, and then of course he comes home and has to unload the effort and stress that took. It can manifest as anger, exasperation, etc. - I try to remind myself what he's going through and just to be there to "receive" these behaviours, rather than becoming upset and trying to course-correct (I'm far from perfect in this). It's up and down, but in the macro trend, he's getting better at these transitions at a remarkable rate.
I view this stage as normal learning and it's something we all still do subconsciously as adults who go out into the "real world" and come home. How many of us are exactly the same in both situations? How many of us feel more relaxed when we get home? Learning to transition from the "outside world" to home and simply relax is a learned skill, IMHO, and hard at first for little ones and they cope with the transitions differently. Honestly, if he behaved the exact same at school and home and handled that transition with ease, I would be combination of worried and blown away.