The UK government isn’t allowed to issue mandatory ID cards after the Identity Documents Act 2010: https://www.legislation.gov.uk/ukpga/2010/40/crossheading/re...
No ID cards are to be issued by the Secretary of State at any time on or after the day on which this Act is passed.
The UK equivalent of a SSN is a National Insurance number, but it’s totally possible to be a citizen without one (e.g. naturalised non-working spouse). NI numbers also aren‘t guaranteed to be unique (if a person is assigned a temporary NI number it’ll be non-unique with other people with the same birth date).