Within Europe there is no such system as European Social Security Number or European Identification Number. A lot of countries have their own system, and other countries are struggling to get a system into place.
The struggle of some countries has to do with historical reasons and with privacy aspects. Unique identifiation is not always used in favour of the community. And some of the used identification systems contain privacy-sensitive information, among others date of birth, gender and/or place of birth, where older systems might even contain religious or other privacy-senitive information.
A wide range of countries use the combination of date of birth, gender identification and the political region where you are born. In such a mechanism it is most common that part of the identification number is a 2-digit or 3-digit serial number to identify the unique male or female born on a specific date (or born on a specific month). Some countries provide odd serial numbers for male, and even for female. Bulgaria is the only one that wants “odd” females. Some countries like to divide on range (0-499 male, 500-999 female). And some countries like Norway make nice combinations to include the century of birth or period of birth in the serial number.
This ‘number’ generation brings the effect that pretty soon you will encounter the maximum number of citizens that the system can handle on a specific day. Some systems run out of numbers if there are more than 500 males or females born on a day. The Denmark system encountered that situation in 2007, where due to immigration the population exceeded the system for January 1st 1965! The Denmark system (CPR-nummer) has a 3-digit serial number where one of the digits is also the control digit (diminishing the possible numbers than from 500 to less than 50).
Remarkable to see what some countries are doing to solve the ‘century’ issue, people with the same ID but born in the 19th, 20th or 21st century, they add 20 or 40 to the month. Same is true for foreigner identification, e.g. Sweden that is adding 60 to the day of birth. Or again Sweden that is adding 20 to the month to distinguish persons from organisations.
If you want to see the details on these systems you might watch http://prezi.com/csnv3cynv4ai/ or http://en.wikipedia.org/wiki/National_identification_number. Be prepared, definitely there have been PhDs around to invent these systems.