It sounds as if you also need to know your National Insurance Number when applying for a code to hire a car. What your National Insurance Number has to do with your driving licence, and why providing your Licence number and postode is not enough, I don't know.

From http://www.moneysavingexpert.com/news/travel/2015/06/hiring-a-car-youll-need-a-dvla-code-to-do-so-fr...

How do I request this code?

From today, the paper part of your licence will no longer be valid, and the traditional paper-only licences may not have your latest points and fines recorded, so you will have to plan ahead and request an access code for the company from the DVLA.

You can do this via the new Share Driving Licence online service at Gov.uk, or by calling the DVLA on 0300 083 0013.

You will need to provide your driving licence number found on your photocard (or on your paper driving licence for paper-only motorists), National Insurance number and the postcode on your driving licence.

The code will be valid for 72 hours, meaning those travelling abroad will have to be mindful of journey times and time differences.

Drivers will also be able to download a PDF of their licence, which will be worth taking as a precaution, but they will still need a code as well.

This 72 hour only for the code to be valid seems as if it could be a headache for someone going abroad who wants to hire a car when they get there but is having a stopover on the way.  If they don't know which hire company they want to use until their arrival and so can't book from UK before they leave, they will have to hope that internet access and the DVLA website doesn't pose any problems when they need to get a code, and they will have to know or write down their National Insurance Number before they go, by the sound of it. .