I would never hand over my ID because put simply agencies are illegal operations and in many cases are run by the type of characters who you really wouldn't want knowing your real address/name/driving licence number.
The idea of an agency asking you to sign a contract is is rather deluded on their part. You never 'work' for an agency, you are a self-employed, independent contractor and are free to walk away at any time. You will get lot's of agencies who try and pretend otherwise but put simply they are talking utter bollocks.
My old agency tried to get me to hand over ID and proof of right to work in the UK at one point. They dressed it up as an attempt to make sure we were all of legal age etc (there had been a case not long before that of an underage girl working for a NE agency). They said they were worried about ending up on the wrong side of the law, however as they were running multiple girls out of flats at the time I think that might have been a case of shutting the door after the horse had bolted. I told them in no uncertain terms to kindly bugger off
No agency I have worked for has ever known more than my first name. I have heard horror stories of ladies having heavies sent to their addresses when they try and leave agencies, ladies getting outed by the agency owners, all sorts of nastiness. All of which could have been avoided if the ladies had better protected their anonymity.
So my advice is and always will be to NEVER hand over your personal details to an agency. If you are asked to then look for another agency to join. Don't be fooled by so called 'up market' agencies, all that takes is a fancy website, glossy pics and a well worded blurb. It doesn't mean for a second that the people running it are reliable or trustworthy, in 99.9% of cases they usually turn out to be quite the opposite.