You can get contactless PAYG prices without any identification in London by using an Oyster card. You can buy them from corner shops without ID, and top them up at the same places with cash.
I doubt this really affords you any extra anonymity though. Tube and rail stations are so heavily covered by CCTV, and the police have many times tracked peoples entire commutes and movement through CCTV only.
To hinder traffic analysis across multiple trips, you should also have meeting where you swap Oystercards with others, similar to how people would (and still do?) swap grocery purchase tracker cards with each other, to get the discount with less surveillance.
You buy paper tickets at kiosk where camera can get close-up view of your face. Cameras can also you track when using the ticket at barrier.
Also, contactless prepaid cards can be anonymous if don’t register them or attach to credit card filling with cash. Can even swap them around with other people.
It's 1970s tech. Even if they had given them unique codes, they had no way of catching duplicates. The magnetic stripe has a very limited amount of storage space and, as far as I know, it's all used for other necessary information.
It also explains why the rail operators are moving to QR codes on paper tickets, which are in every respect worse. (They are absurdly large tickets and take much longer to scan at the barrier, creating queues.)
I would assume contactless payments are easier to surveil compared to paper tickets, similar to cash vs credit card payments.