No, I haven't explained it well either - because I realise that I had managed to imply that a Traveller is identifiable from their clothes, which isn't true in the slightest.
What my experience was, was that the school had an 'identity as a community' - a character, a set of values, an ethos, a way of being, much more than a set of rules or a single building - that was greater than the sum of all the individuals within it, even though all of those individual identities were preserved and valued (this doesn't happen in all schools, i will entirely agree at this point. But in this school it did - value for the individual but also the creation of a strong and inclusive community identity).
One of the 'tricks' that the school managed to pull off very successfully was in bringing together many very disparate 'out of school' communities and making them all feel part of the school one - and I cite the Travellers in particular because they were a community which lived very separate lives and because all previous generations had been unschooled but all our local traveller children were now attending school.
The uniform was a symbol of this - and only a symbol, it wouldn't have happened if there hadn't been more to it than just the uniform.
I suppose what I'm saying was that uniform isn't sufficient on its own to make someone part of a new community BUT if the new community is exceptionally good at welcoming people in, then the uniform can become a recognised 'shorthand' for 'you've joined us'. A bit like you can be a Scout without a necker or woggle but acquiring these has become a shorthand for 'you've joined'.
It's not about removing Traveller or settled identity, in any way. it's about creating, in school, an identity of 'belonging to this school' which does not in any way remove any other identity but is a new and separate 'group' identity. It's precisely because none of the individual identitities that you mention - nationality, religion, cultural - are solely defined by clothing that the 'transceding' eidentity of belonging to the school leaves those unaffected while also creating an inclusive group identity. Which happens to be 'those who wear the sweatshirt', but that is simply a symbolic shorthand for 'those who belong in this school community, wherein all are welcomed'.
Of course, you can achieve a community without any visible symbol of belonging. But a visible symbol can have an instantly transformative effect in some circumstances.