As I read it, the statute means a license broader than an FFL; it would include a license to purchase a firearm issued by a governmental unit or an CCL provided the governmental unit "verify that the individual is qualified under law to receive the license."
I did a quick search and found one published case of interest on this point, U.S. v. Tait
, 202 F.3d 1320 (11th Cir. 2000) in which the court said that Alabama's licensing system for concealed carry met this requirement even though the sheriff did not do criminal background checks and Alabama law (at that time) said a sheriff could issue a license "if it appears that the applicant has good reason to fear injury to his person or property or has any other proper reason for carrying a pistol, and that he is a suitable person to be so licensed."
case is available publicly at http://openjurist.org/202/f3d/1320/united-states-v-tait
with the pertinent portion beginning at paragraph 17. I'll also remind those viewing this message the Tait
case is only the law in the 11th Circuit.