Why would a nurse's license number be a matter of public record?

Because they are documents that are issued by the state, and the state's business is the people's business. All such professional licenses to practice are required by law to be posted in a public area accessible to the pubic in the location where the licensee conducts their profession. There is nothing confidential about the serial number or the name on the license being public knowledge, but the home address and contact information of the license holder IS confidential.