The BLS is the bureau within the labor department that computes these numbers. The BLS is the department of labor.

  There are different numbers based on unemployment claims vs. a survey of people who say they're looking for work.  It is possible to look for work while not being eligible for unemployment insurance, for instance.

During COVID in some places it is possible to receive benefits while not actively looking for work, as the government has shut down many jobs. This is a matter of nuances in the definition of the figures.