@Jim75
After the fact, I'd certainly say yes.
The full name of your guest is on the reservation, but you are allowed to ask him for full names of all of the other guests.
Tell him there is ID check at check in.
I'm sure you have a right to do this in your part of the world. Just handle it in a cool way, and if he's not okay with it, then that would raise a red flag with me.
I have to do this as part of my city's regulations and I write down the ID #s. Hotels and any other commercial rentals do that too.
The pleasant side effect is that guests somehow take more notice of the fact that they are now registered.