Scares me. Once obtained, this info will forever be in their device(s) and could become public at anytime, with or without their permission. Doesn't take much, sometimes, for people you don't know to turn on you. I have no interest in knowing the personal info of the women I see except for a couple with whom I became friends with over time. Otherwise, it's best, IMHO, if it remains an anonymous transaction. (Elliot Spitzer, et. al. ring a bell?) The reason many use hobby phones and meet away from home. Heck, not only do we use fake names on TER, but we use aliases to disguise our fake  names.  Just my thoughts as you askef.

I can understand if it makes her feel safer, but it makes me more nervous.

I don't mind using my real first name when making an appointment. If she wants a last name, mine is Green.
I don't think I've ever given my real last name.

It says it, but I bet it wouldn't be honored.

My thoughts exactly...extinguished

Typically when they want a full name they also want something else such as a look at your id, Linked in or Facebook account, etc.  That complicates things.