If you store it just as a string it means that you cannot do anything useful with it like age-dependent logic or you just pass on parsing logic to users of the field.

FHIR in my opinion has a pretty good system for dates (including birthdates): YYYY, YYYY-MM, or YYYY-MM-DD. (Not knowing your exact birthday is common for some countries).

https://build.fhir.org/datatypes.html#date