It's somewhat unusual but certainly not unique. Wikipedia maintains a list of examples on voter identification laws. I'll quote a bit from that. I'll not cite those that require ID because most do. I will cite some exceptions, and as you can see the rules vary a lot on a case by case basis.
Countries not requiring some form of physical ID or specific election-related documents
No form of ID is required to cast a ballot at an election; instead, voters are asked three questions before being issued a ballot, so that they can be checked off the electoral roll: (1) what is your full name; (2) where do you live; and (3) have you voted before in this election?
In Swiss cantons (i.e. the subnational political level in Switzerland) that still use the Landsgemeinde or cantonal assembly; Historically, or in Appenzell until the admission of women, the only proof of citizenship necessary for men to enter the voting area was to show their ceremonial sword or Swiss military sidearm (bayonet); this gave proof that you were a freeman allowed to bear arms and to vote. Women, and men who choose to do so, may show their voting card instead.
The United Kingdom
There is currently no requirement to have identification to vote in elections in England, Scotland and Wales, before any election all eligible voters are sent a Poll card by their local authority although it is not a requirement to be in possession of a Poll card to vote
Some states in the United States, according to a different Wikipedia page:
No ID required to vote at ballot box: California, Illinois, Maine, Maryland, Massachusetts, Minnesota, Nebraska, Nevada, New Jersey, New Mexico, New York, Oregon, Pennsylvania, Vermont, Wyoming, and Washington, D.C.
ID not (strictly) required, but some specific election-related documents are
Germany uses a community-based resident registration system. Everyone eligible to vote receives a personal polling notification by mail, some weeks before the election. The notification indicates the voter's precinct polling station. Voters must present their polling notification and if asked a piece of photo ID (identity card (compulsory in Germany), passport, form of identification). As a rule identification is not required other than by the polling notification. If the voter cannot present the notification, a valid ID and an entry in the register of voters can qualify for voting.
The INE elector's card is currently used in Mexico as the main mean of age and identity validation for legal, commercial and financial purposes, making this a vital document for all Mexicans over the age of 18, and consequently broadening the chance for more citizens participating on election day.