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All it takes is one breach or vulnerability and then yes, they DO have access to all your data.No they don't. If they breach the health system, they don't have access to tax returns.
Just because people are identified by a single ID number doesn't mean all their data is being stored on the same server. And for purely organizational reasons, that's incredibly unlikely to happen.
And I don't know what you mean by "steal your identity". People's names are date of birth are generally a pretty unique identifier already. It doesn't really matter if systems use that or a single ID number to identify you, or if hackers look you up by your name.