It's been federal law since 2003:
FEDERAL LAW
There are a number of federal laws that require an individual to provide a SSN. The most similar law to CGS § 4a-78 is 42 USC § 666(a)(13), which lists several procedures every state must have to improve the effectiveness of child support enforcement. It requires that a SSN be provided on:
1. any application for a professional license, driver's license, occupational license, recreational license, or marriage license;
MT also has it on their website an SSN is required to buy a fishing/hunting license. Unsure why you would be an exception.