So you setup something like "account 123 can debit up to 100 EUR monthly" and the bank enforces that.
You could theorically withdraw money from an account by faking the procuration, providing you know the id details of the account owner, but you are still required to present your id card which contains a digital photo of you to prove you are who you pretent to be.
Fake id are maybe common in the us thanks to your broken SSN system but in europe, digital id cards are centralised from the moment you're registered as human being at your birth and banks have access to that system when they check your id.
Any attempts to withdraw money from your account leave a trace, and in case of fraud, you would be quickly with the cops on your trail.
The UK and Denmark don't have identity cards. At least in those countries, there is occasional fraud. The banks will fix it (at least in the case you describe), but it can happen.
This isn't true at all. The direct debit system allows any participant to withdraw money from your account, providing just your account number and a claim that you authorised them to do that. You never have to directly tell the bank anything, and in many cases you can't even opt out of the process or deny access to an individual withdrawer through the bank.
The banks are meant to limit access to the direct debit program to companies they trust, and cut off access to bad actors, and make whole customers who were debited in error or by fraud. How enthusiastically they do this depends on the bank and the local regulator.