Brazil
In Brazil there are two different systems. The first one, the Registro Geral (RG) is a number associated to the official ID card. Oddly, although the ID cards are supposedly national, the RG numbers are assigned by the states and a few other organizations, such as the armed forces. So, not only is it possible for one to have the same RG number as a person from other state (which is usually dealt with by specifying the state which issued the ID card), but it is also possible to (legally) have more than one RG, from different states.
The other system, the Cadastro de Pessoas Físicas (CPF) is federal and supposedly unique (barring fraud), but it was created originally only for purposes of taxation (a related system is used for companies, which is called Cadastro Nacional de Pessoas Jurídicas -CNPJ). One, the other or both numbers are required for many common tasks in Brazil, such as opening bank accounts or getting a driver's license. Generally speaking, the RG system is more widespread (many poor people, who have no bank accounts, never get around to registering for a CPF), but its practical shortcomings have led to debate about merging both systems into a new one, which would be based around the CPF, so they say.
Another type of registration is the Social Security Number, which is originated when a person creates one in the National Social Security Institute's site or starts to work for a company, when this one has to register the employee in the Social Integration Program.
Read more about this topic: National Identification Number