"member" or "subscriber" number are not appropriate because a person is neither a member nor a subscriber to the national health service.
Even in the case of private healthcare, it would be something like "customer/patient number" or "policy [holder] number".

The penultimate example here uses "Policy #".
https://www.template.net/business/discharge-summary-template...

Here's an example of "Social security number" in the US.
https://www.addictionary.org/templates/fake-hospital-dischar...



4 Erasmus + Programme – VETPRO Study Visit
Public healthcare in Spain You must register with social security to get a social security number. You’ll need to show your passport or ID card, residency certificate and a completed application form. You’ll also need to have registered your details (address etc.) at your town hall. Once you have registered with the TGSS you’ll be given a social security number and a certificate stating that you’re entitled to medical help.
https://slideplayer.com/slide/13443332/


If you’re including medical evidence, staple it to the form. Make sure you include your name and National Insurance number on every separate sheet of paper.
https://www.citizensadvice.org.uk/benefits/sick-or-disabled-...