Sadly the solution isn't banning Nikes or whatever the current "they're being petty about it" thing is - they'll just find something else to latch onto.
That was my point indeed. Although thinking about it again, in some countries (not mine) they got school uniforms for this reason. I don't know if it helps (as in, there will still be bullying no doubt, but is there less bullying without than with school uniforms?)
I suspect if there are variations on "amount of bullying" it comes down to culture rather than "school uniforms" - at least one Japanese manga is all about bullying in a school uniformed school, so it must happen enough to be a bit of a trope.
It may happen less in places where the private schools can be school uniform or not, but there I suspect it has more to do with expelling the bullies than anything to do with the uniform itself, since it won't usually cover other status symbols that can be obtained (shoes, hair decorations, etc).
That's not to discount that the uniform may be useful for various reasons.
The solution is for mature adults to not behave like children because they understand being petty about brands creates serious problems both for themselves and everyone around them.
Children doing it is just part of the series of learning experiences that comprises maturation.