if they just look at it, and hand it back to you, you shouldn't have anything to worry about.

if they make a copy of it, then you might have a reason since that means they are creating a paper trail.

It may be perfectly safe, but do you want to take that chance?  As Naomi said, did they actually photocopy it, or just look at it and hand it back to you?  If they intend on copying it, I would find another agency.  If your identity is discovered in an LE raid, and that's all that's in their records, LE cannot prove you did anything illegal.  However, they might publish a list of names they discovered which could prove to be detrimental to you, especially if you are married.

I would not give them my license.  There are plenty of agencies out there to choose from.  Unless you are very new at this and have no references yet, I would stick with indies.  In the 6 years I have been hobbying, I have never used an agency.

Swim

If a provider screens a certain name & info, some providers require you to show ID upon arrival to prove that you are the same person she screened.   I've never done it but I can see how some might desire this.
Make a copy?  No friggin way!!  Paper trail.  Identity theft.  Blackmail?  Extortion?  NOPE, NOT doing that.  

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HV

I use agencies. They are fine. Like everything else you have to find a good one that will be discrete. They don't need a copy of your license. I wouldn't deal with an agency that does this. They can discretely screen you and you should be OK to use their service from then on without any problems.

My advice, find a different agency. Don't give them a copy of anything. The provider may ask to see an ID when you arrive to be sure it matches what you told them about who you are but they don't need to keep anything.

Some agencies, and even some independents, will ask you to 'show' your id, as part of screening. Many guys are uncomfortable with that, and will not show it, and that is fine. They will just have to see providers who don't ask to see ID. Some guys are comfortable with showing their ID to a reputable gal. It is up to you where your comfort level is, as far as showing ID or not. For me personally, I have never been asked, but I would gladly 'show' my ID to certain ladies if I trust them. However, I would never, ever let anyone make a copy, or give them a copy of my ID.

Showing your ID, and someone copying your ID, is definitely two different things! and there are reasons other than the police getting a warrant. (ID theft, etc.)

On my first visit with a provider it was with a Dayton agency.  The girl asked to see 2 forms of ID with my name on it.  She glanced at them and handed them back to me.  This agency is really, really big on safety and security for the girls, so looking back, it doesn't surprise me.