I think as a set of compromises it would seem reasonable to me to go with:
School branded ties and sports socks (generally don't need multiples, fairly cheap, and don't usually need regular replacement) plus sew on badges for either blazers or jumpers - again fairly cheap and you can unpick and resew onto bigger sizes if required.
Then generic (my school used to specify no visible branding to avoid pushing costs up) shirts, day socks, trousers, shorts, skirts, jumpers and blazers (to have badges added to these two), polo shirts and shorts etc for PE.
That way you can still facilitate people being smart etc. Maybe they could even sew their own badges onto their blazers for their first year 7 DT lesson if the school wanted to be really equitable.
Stuff with striped school shirts or tartan skirts etc just seems absurd to me and unnecessary. You can be perfectly smart with the suggestions above.