the most relevant thing I can find on the fsa website is this, which relates to traceability of gmo crops.
"For example, where production starts with a genetically modified crop, the company selling the crop for feed production would have to inform any purchaser that it is genetically modified. Information must be retained for five years."
I can find nothing on traceability of final purchasers of such feed, and gm crops are traced more than the rest.
But it's all under EU legislation, which means mountains of impenetrable regulation in bureau-speak, so I guess it is possible. But I would still ask the shop to prove it.
Shop staff like Asda's may be on standing orders to try to get as many customers' data as possible and even incentivised to do it. If it is law then the relevant law should be displayed where they are asking for it ie at the till.