It looks like the Guardian is basing this off reports from Dispatches and The Times and iy seems that it's really mot quite clear how it works. Here's the de-paywalled Times report: https://archive.is/s5eDe

... a “clean room” can match specific shoppers, or small groups of shoppers, with specific television viewers and work out when they are likely to be the same person.

Few experts are willing to reveal the exact science, but the software can make a remarkably accurate match. This is because supermarket shoppers reveal so much about their income, lifestyle, location and family set-up from what they put in their baskets — and because television viewers expose so much about their habits, income and location from what they watch.

For example, the experts will know when someone has stopped watching I’m a Celebrity, but still watches Coronation Street, and where they log on to watch it. “With data matching, they tend to use ‘lookalike data’ — this person [on a supermarket database] looks very much like the same person [on a broadcaster’s database],” explains Duff.

They can also add in third party databases, such as Facebook, and the matching is uncannily accurate, as well as being privacy compliant, according to experts.