Re: This requirement for paper bills/statements...
"Never tried to get onto a client site?
Never tried to borrow an item of equipment?
Never tried to book tickets to an event?"
These are all valid reasons for requesting to see your ID - not to keep it. Crucially, they're all events initiated by you; so you're not 'being continuously stopped and asked to show them', as you originally wrote.
Additionally your use of the word 'gotten' leads me to believe you're a USAian with an outdated and stereotypical view of the EU based on a book you once read called "Europe: There Be Dragons" rather than any actual personal experience.