We had this exact same issue around the same age. Despite a pretty generous maintenance payment each month the children used to turn up at ours with not so much as a toothbrush when I met DP, once DSS was even sent on a sub zero night such as tonight with trousers that were swinging halfway around his calves and no coat.
So every weekend DP had been buying clothes, pants, socks etc that never saw the light of his house ever again.
The way we resolved it was to buy them both a cheap overnight case, like the on-plane cases you can get and a wash bag each, and we told them that whatever they wanted to have at ours and mums needed to go in the case when they went between the houses.
They both loved the concept of packing up everything they wanted, I don't think mum could kick up a fuss because each case was theirs and mobile with them, and it meant we didn't have to buy them two new outfits every weekend. AND when they were younger they could bring toys, teddies etc that they were attached to at each home with them.
Now the two of them tend to be pretty good at doing this as standard. We rarely see anything that mum has bought them, but the nice bits we bought them that they favour do make it back to ours. We're not buying them designer gear but DSD has a couple of Hype hoodies and some Zara skinny jeans that he's attached to. The rest is all cheap t-shirts and trannies bottoms. When we bought him trainers at Christmas he refused to take them home because he wanted to keep them at ours "for best"
DSD takes everything she needs as standard, including half of boots as a make up bag and all her undies etc. We've bought them both fluffy towels and dressing gowns to have at ours as their own but PJ's etc are still mobile.
DSS loses pants and socks (they just don't come on his radar to pack) and funnily enough mum never volunteers that she's stockpiling more than she needs at hers. But cheap pants and socks are a marginal cost compared to what we were spending before.
I will say that it doesn't hurt when they are the age to appreciate it to buy them something a smidge more expensive that they'll want to wear EVERYWHERE. Not suggesting for a second that they're decked out in designer gear but I got a Hilfiger T shirt for DSS in a sale and DSD her first pair of Topshop jeans. Both items seem to make it back to ours all the time along with the items that go with them