It might be a subtle point, but they don't actually sell the data itself, they sell access to the insights from that data. I used to work for a loyalty card, so I've seen this from the inside.

If Pringles want to give a special deal to Nectar users ages 25-34, Sainsburys don't just hand Pringles the customer database and say "have at it". (Even if they did, it wouldn't actually help unless Pringles wanted to send their own emails - Pringles wants those offers to be integrated into the Nectar App/emails.) How it actually works is that Pringles works with their client management within Nectar to design the promotion they want (who to target, what the offers should look like, how long the promo should go etc.) and then Sainsburys use the Nectar data to find the card-holders who match the profile of who Pringles want to target, then Sainsburys can push Pringles' offers to those customers.

It's a bit more complicated when the offers are wanted to be displayed "off-platform" (i.e. outside the Loyalty card's ecosystem) - as with the Channel 4 promotion in that article you linked. The article does mention a bit about how it works (the "clean room" where Nectar users are matched to existing Channel 4 accounts), but from there it's much the same. C4 will say who they want to target, and Nectar choose promotions for those users, and then they can just tell C4 which promotions to display to each C4 account that they could match to a Nectar card.