I’m hoping someone can help me find some demographic data for the US. I'm looking for data with first name, last name, sex, date of birth, and zip code and a unique identifier for each person. I want to calculate the rate of people in the population who have the same first name, last name, sex, date of birth, and zip code but are unique individuals.
Since this was tagged as a US Census Question - the Census Bureau does not publish data for individuals - it's against the law: https://www.census.gov/history/www/reference/privacy_confidentiality/ - individual responses are kept confidential for 72 years before they are released. For current data you have a choice of summary data (responses summarized by geography) that's accessible via the American Factfinder (on the Census Bureau's website) or microdata - samples of individual responses with personal identifying information removed. You wouldn't be able to get names or birthdays - just age and gender. The best source for microdata is probably IPUMS USA. The summary data is provided for small statistical areas (census blocks) while the microdata can only be summarized for large areas.
The US post office also does not provide lists of individuals or customers - only standardized addresses that can be used for mass-mailing. You would probably have to purchase or get access to a proprietary database for this type of information (like the US Consumer / Lifestyle module in the ReferenceUSA database, published by Infogroup).
This resource is available for the State of Ohio.