Is there a legal way to verify one's US citizenship, for example, comparing name & social security number to an official database?

Yes, you may verfiy social security numbers and if you employ people you should be

It is a free service for busienss though the social security web site

Social Security Online - Business Services Online

I understand that the SS admin can verify validity of a soc sec number, but how can I verify that a person is actually a US Citizen?