In BrE, if you are stopped by the police and asked to tell them your name, we would say "The police took down his ID" - here, "to take down" means to write on a piece of paper or in his police notepad.
If I heard "The police took my ID", I would assume that they policeman had taken possession of an actual identity document - passport etc. That would not happen in the UK unless you were actually under arrest.
Note that in the UK, we are not required to carry any form of identification with us by law. If we are stopped while driving a car (stopped for a legitimate reason - we don't have random stop and search here), we may be asked to say whether it is our personal car, give the registration number and then give our name and address. A check can be made on a mobile computer to see if the car's registration number and the name and address of the owner match what we told them. If we are found to have committed a minor offence (ie one of our brake lights isn't working), we can be told to get it fixed and also to attend a local police station with our driving licence and the ownership documents of the car within 7 days.