In most of the EU, IDs are issued for free to everyone above age 14/15 (and in many countries you can get one even for a newborn for a small fee). Since everyone has this ID, all banks (nearly?, I haven't seen one that doesn't) require ID card and/or a passport to open an account. For medical care you have a separate card with your compulsory medical insurance information that you present to the doctor but in the worst case they can just look up the info using your ID/passport.