They are not worried that someone is going to come in, and steal your appointment. They are worried that someone with the same name as you might show up on the same day and the doctor might treat the wrong patient with the wrong information.
This is an completely different risk profile than a form on the internet.
I have the same name as my father (first and last, , different middle). We live at the same address. It’s a small town so we share a lot of the same doctors. We use the same pharmacy.
For just a bit of extra spice are birthdays are only two days apart.
A person she hadn't seen or talked to in 20 years had moved to this town neither of them were from and named their kid the same name.
Because I have literally seen this go wrong: “Mr John Smith, you’re here for procedure X, yes?” “Yes” Some other provider overhears: “I thought that was Mr Jones for procedure Y” “Are you Mr smith or Mr Jones?” “Mr Jones” “Then why did you say yes when I asked if you were Mr Smith” “I assumed you knew best”…
People do weird things in healthcare settings