School uniforms don't indicate nearly the same level of subordination to a command structure.
School uniforms don't really deal with this dynamic because they're only relevant for interactions internal to the school rather than interactions between the school and the outside world.
Well they also have an effect on the interaction with the outside world: remember that Trainspotting scene when Renton wakes up to find out that the girl he picked up is a schoolgirl?
My school had explicit rules about how we should behave when in uniform outside of school, and we lost some agency because of that. Although I have to say it also felt empowering, to some extent, that we were kind of ambassadors for the school. We would definitely show off sometimes.
Similarly, sweet shops near another school I attended had signs saying 'no more than 3 children at once' or something similar. The uniform part was implied but I think a parent taking in four kids on a weekend day would have a different reaction from the shopkeeper than four uniformed children at 3:30pm.
Similarly, a group of teenagers in uniform may be less threatening on the street than one out of uniform, especially as fashions change. A certain school's uniform communicates a level of wealth or manners (or lack thereof) and therefore sets expectations for behaviour. A child alone in the library working in uniform may be far less likely to be approached by concerned staff than one in mufti - at least this was the case when I frequently stayed at the library outside of school hours.