Fire is a plasma, not a solid. Of course it won't make a shadow like the match and fingers. Think more along the lines of shining a light through a thin bit of liquid or gas. There may be a slight distortion, but barely visible.
Light sources can be opaque and therefore cast shadows if there's a significantly brighter light source behind. Fire being a plasma, it's actually quite good at conducting electricity and therefore attenuating the light that tries to go through it: fire absorbs light. If it absorbs more than it emits, you'll see a shadow. It's basic radiative transfer. Let Lb be the background irradiance, Lf that of the fire and A the absorption coefficient of the flame; without flame you see Lb, with flame you see (1-A)Lb+Lf, which can be lower than Lb and therefore appear as a shadow.
Too bad the average population of this site is stupid as fuck. If you posted a photoshopped picture of a dog with a cat head, they would believe that it was real.
Light
The visible reminder of Invisible Light.