Welcome to ribnet!
Do you mean the "chassis number" (VIN in car speak) or the towing vehicle's registration plate?
It's a Euro wide thing that you will need the towing vehicle's registration plate, country ID plate and suitable lighting on the back. Thankfully harmonised Europe means if it doesn't have a lighting board whne you pick it up a quick diversion to a car / trailer shop will get you one. (or better still - ask the seller and if not bring your own!)
Regarding the trailer's chassis number I'll need to hand you over to the Irish & Dutch guys who will know the rules for the respective countries better than me, although I guess whether it even needs one or not will depend on the age of the trailer.
What's the new boat?