I've tried to research this topic but all answers boil down to the fact that the directive (Directive 2006/126/EC of the European Parliament and of the Council of 20 December 2006 on driving licences) requires it. No answers are given to why the directive requires place of birth to be displayed on the license.
The document 2003/0252 (COD) provides background discussion of rationale for an earlier version of the regulation. Two important considerations regarding the EU licensing scheme are underscored: the desideratum of unique identification, and fraud prevention. The rationale relates to safety – they want to know whether a specific person has been banned from driving for safety reasons, and they are aware that people can make fraudulent claims ("That's not me, that's someone else"). Remember that the right to drive is not absolute, it can be revoked.
An earlier (earliest?) form of the proposal is Document 51988PC0705, submitted 9 Dec. 1988. where the Council of European Communities which contains the skeleton of the present law. Annex 1 gives the earlier required information. The required information is more minimalist, including Surname, other names, place and date of birth, permanent place of residence, and other information on the issuance of the license (where, when valid, who issued). Given this sparse required information, name, place and date of birth will generally uniquely identify the license holder.
It is true that unique identification can be accomplished in various ways, but the reason why the current rule is what it is is because nobody has persuasively argued for an alternative rule that does not include date and place of birth, and date of birth would be independently necessary w.r.t. rules governing minimum age of drivers. Since the requirement was present in 1988, it was just copied from version to version over the decades.
Name and date of birth are not sufficiently unique to identify a person. While name, date of birth, and place of birth do not have to be unique, either, it reduces the number of false positives. Also, date of birth is somewhat better known than place of birth for most people.
Motorist: "Sorry, officer, I seem to have lost my wallet. Everything was in there, ID, license, ..."
Cop: "Tell me your name, date and place of birth, and I'll run a query if you do have a license."