It's pretty common - in fact some will ask for your employment info as well in order to verify you. It's up to you to decide how much you're willing to divulge but do your homework first and check their reviews.

Note that if you go the Date-check or P411 route it's the same deal - they'll ask you quite a bit of personal info so that they can verify you. They don't release that info to the provider though unless you put it in your public profile where she's free to read it. If you're concerned about giving out your full name you may want to consider going the verification service route and that way you only really have to do it once. Be aware that as a newbie they may still ask for other info besides what comes from the service but they're less likely to do so.

I have never used an agency that didn't want my full name.  Most of the time the lady is going to want to see your ID when you arrive as well so don't try giving them a fake name.

Agencies are no different than indies.  Stick with agencies that have a good rep and well reviewed ladies.  Is it 100% safe? Well, no, but nothing in the hobby is.  We all have to decide what level of risk we are willing to undertake and accept the fact that our comfort level may limit the ladies that will see us.  It's just that simple.

An agency would have good cause to know your (provider) legal given name...  & see your ID.  If you are underage, it could have serious legal consequences for them.  I would be glad that the agency strives to keep adult industry all adult.
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In my opinion you should only disclose what you are comfortable with.  Agencies are different than independents though.  Maybe seek an independent with reviews that will verify you in a way you are comfortable.  Happy hunting.
xoxo