This behaviour is just because their IT system doesn’t allow regular users to search for names, just for birth dates. Then they pick you by name from a list of people with that birthday.

This is nowhere near the only use of the "birthday + name == all info" hack in the US medical industry. It's basically one big giant frat club with shakes and implicit trust all around. Except that it doesn't actually work; you can fake being a doctor to just about any US medical office and get nearly any American's private medical data.

Names are not unique