The study's methods and definitions are described on page 10 of the full report, which can be downloaded PDF from their website: https://www.jrf.org.uk/deep-poverty-and-destitution/destitut...
It boils down to whether people can afford basic essentials like shelter, heating, lighting, and clothing. They do check their income levels. Maybe some people would lie about this, but I don't know why they would.
We know the prices of shelter, heating, lighting and clothing - and we know how much money people make. I see no need in self-id here. The equivalent of these surveys in the US place some people making six-figures in the food insecure bucket.
In the UK at least, we don't generally know how much money people make, nor how much of it goes to debts or other dependencies. Income below £10k does not have to be declared.