We have multiple numbers. We have a national insurance number, which is issued by the Department for Work and Pensions, which is used by them primarily. We have unique tax reference numbers used by the taxation department, Her Majesty's Revenue and Customs. We have NHS, National Health Service numbers, and most other departments do have their own unique identifiers for people.
Going back to my original comment, that needn't necessarily be a problem, because there is no reason you couldn't have a number, as in Estonia, that potentially is a super-set of those to enable me to prove who I am to each of those different indexing systems, if you like, but without necessarily their being able to see across my proper identity file.