No, UK not only has no ID card but also doesn't have the equivalent of an SSN. The National Insurance number serves the same primary purpose as SSN, for associating tax payments, but it doesn't otherwise serve as a unique identifier for each citizen - it's not used on credit card applications for example. UK has lower %age of drivers than US due to strong public transport, so mandating a drivers license for ID isn't feasible.

The UK just bought in new ID rules for voting which bizarrely allow for things like a senior citizen rail card to be used. (it's being brought in by our conservative government for same reason as US - to disenfranchise certain voter blocks - hence why a senior citizen rail card is fine but a youth or normal adult one is not)