The relevant provision is Regulation 50(2) of the Child Maintenance Calculation Regulations 2012. Paraphrasing, it says that if both parents maintain a home for the child, and both have equal shared care, them neither can be treated as the non resident parent for the purposes of a maintenance application to the CMS.
The key thing here is 'equal shared care'. That's not necessarily the same as 50/50 - it also takes into account things like involvement in schooling and medical care. It's an area of evolving case law, but broadly yes - if you both provide equal shared care, then he doesn't have to pay. Income is irrelevant.
It exists for a good reason. It's about fairness, and ensuring that both parents contribute to the costs of the children, rather than one parent meeting costs while the children are with them, and also while the children are with the other parent. But where there is a large disparity in income, it risks putting the children in a situation where they have a significantly lower standard of living in one home than the other.
To that end, it doesn't preclude the higher earner from voluntarily paying something. That's what I do - my ex and her new husband are both lower earners, whereas I am not. So I voluntarily pay her some maintenance to bridge the gap. I also take care of the school uniform costs at both houses, and tend to cover the costs of other major expenditures (school trips etc) as well. But it is voluntary, and it can't be enforced. Not gonna lie - there are people who think I'm nuts given how much more money she got in the divorce than me, but it still feels like the right thing to do 🙂.
So, if it is truly equal shared care, then he is right - and your best bet is to talk to him about how you work together on costs to ensure the children never go without. If it isn't truly equal (as you suggest it might not be), then he is wrong and he still has to pay.