In the Netherlands you must always have an ID card with you outside of your house. All persons 12 years and older have to have some sort of ID with them.
Several different items may be used as ID, passport and driving license amongst others. But not you bus card. The driving license is the most normal. It has a number called BSN (Burger Service Nummer, citizens service number) which all government departments, hospitals (gives the hospital access to your national medical record), and police accept.
Whenever I go out I always take my driving license and my bank card with me (as minimum, sometimes I even put my trousers on). I've never experienced it as intruding on my privacy, it gives me instant access to many services as well as being able to drive my car and anybody else's car with permission. That permission means that you are then covered by the car insurance of the car owner. That can be very handy.