They will have a record of times paid for against vehicle registration numbers, which were presumably entered on the machine at time of parking. You had a ticket which said the same thing.
The driver entered into a contract by reading the signs and parking, and their notice is now an invoice for payment under that contract. The liability rests with the driver, who can be identified by the keeper if wanted, but if not the liability can be transferred to the registered keeper under the stipulations of the Protection of Freedoms Act 2012.
If not paid, the driver/RK can be taken to court for breach of contract for up to six years.
There may be technical failings with their notice which fail the requirements of PoFA 2012, but they will dismiss any appeal based on this and it's likely a county court claim will eventually happen.
Most of these claims are discontinued eventually if defended, but we're talking about a period of several months to a year before that happens.