This comes from my personal experience.
I went to a relatively elite private school in South Tangerang City, Indonesia. The student that went there were the rich kids, I on the other hand wasn't rich but my parents were wealthy enough to afford my education at that school.
Anyways, That school actually had uniforms but every Thursday is a free dressing day. THIS would be the chance for the kids to show off their expensive clothing. We were all kids but we wore expensive clothes and clothes that went over the top sometimes. One kid who was in 1st grade even went as far as wearing a snow white dress to school (I kid you not, Cringed HARD when I saw it)
Anyways, What I noticed from there was that a lot of the students made groups based on how "wealthy" they looked. Every Thursdays, I'd only wear jeans, A blue T-shirt, My hair in a pony tail and wore pink square glasses. Yep, I dressed like your typical nerd you see in movies and I was an outcast.
The reason I believe that uniforms would create unity is because studies show that we are more likely to be attracted by things that look like us. So wearing the same clothes as everyone else would make people feel more comfortable around one another.
I also believe that it'll avoid competition in popularity based on wealth. Like I said before, A lot of the students in my school made a friend group based on wealth. Even though they could still show off by using the latest technologies or wearing expensive accessories (It wasn't allowed in my school, But I bet a lot of other schools are okay with accessories) by making them wear the same outfit it would AT LEAST reduce the "showing off"
The students in public schools should be able to wear any clothes they like if students have a choice of fashion it will open their minds up to what people like and being more creative towards them if they´re able to choose what they wear. Although if they are in a private school they should have a uniform