Don’t think of emissivity as a fundamental material property in the same way as other physical properties.
It is actually a little complicated since it depends on the surface of a material as well as what the material is made of.
A smooth polished metal surface will have a lower emissivity than a rough or textured metal surface. Usually a rough oxidized surface will have a higher emissivity. The emissivity can also depend on the angle and wavelength especially for textured surfaces.
Thus you measure the emission from the smooth metal surface with a radiation thermometer it will appear to have a lower temperature than the rough surface. For the same temperature if you point the radiation thermometer at a black body of the same temperature of a metal with a low emissivity, it will read a higher temperature from the black body.
if you have a heat transfer calculation the emissivity is a useful parameter in your equations. In general radiation thermometers is calibrated by the black body radiation.
Note that like mentioned in another answer. The black body spectrum is the radiation emitted as a function of wavelength and when people talk about it they are generally referring to the general shape of the spectrum as a broad band light source who’s peak will shift with temperature.
So when you heat up the tungsten filament and look at the white light spectrum from a light bulb it will have the same shape as other black body radiators at the same temperature.
But if you have a polished tungsten plate at some temperature and an oxidized copper plate at the same temperature, you would find the the copper plate would have a higher emissivity if you measured the temperatures with a radiation thermometer calibrated to a true black body source.