I think the equivalent in the UK is the NI (National Insurance) number. We get a plastic card with it on (which I keep in my wallet).
I don't know it off by heart largely because the only times I need it are when starting a new job or changing my GP. I think I've had to write it down less than eight times in my life!
Interesting fact for fellow Brits: The mapping between NI numbers and people isn't one-to-one or even one-to-many. It's many-to-many. There are NI numbers that are assigned to multiple people, and some people have multiple NI numbers assigned to them. The combination of NI number and full name is supposedly unique, but there's no system in place to ensure that. If I were to find someone else with the same NI number as me and change my name to match theirs, we would be indistinguishable to the government (for tax purposes anyway).