"HMRC say they will reclaim Gift Aid from the donor if not enough income tax had been paid to offset it. That implies they correlate a charity's claims with the donor's tax records .... "
It does, but they don't. All the charity collects for a Gift Aid claimant is their name, address and the date of their declaration. No DoB, NI number or wany other easy way of linking to a tax account. In fact I've long thought that HMRC seem to have no interest in policing the scheme.