The paper is talking about effective temperature for specific photon energies. Additionally black body emission refers to thermal equilibrium conditions which the LED clearly is not.
And then there's the near-field coupling they're using:
> Recent experimental advances in near-field radiation have shown that heat-transfer rates on the nanoscale can exceed the blackbody limit by several orders of magnitude14–17, owing to contributions from evanescent and surface modes18,19. As a result, energy conversion rates can be greatly enhanced on this scale20.
So this whole system relies on not being a black body on several levels.
> Recent experimental advances in near-field radiation have shown that heat-transfer rates on the nanoscale can exceed the blackbody limit by several orders of magnitude14–17, owing to contributions from evanescent and surface modes18,19. As a result, energy conversion rates can be greatly enhanced on this scale20.
So this whole system relies on not being a black body on several levels.