Operational amplifier bandwidth - why does manufacturer tell it's DC when it is not
This isn't an opamp (operational amplifier). It isn't even a differential amplifier. This is a radio frequency attenuator and amplifier on a chip.  



  Why would they say it's DC when it's effectively not.  


Because the inputs of the amplifier and attenuator are DC coupled to the respective outputs.  



It could theoretically be used for amplifying a signal with a DC component, although in most (probably all) applications you wouldn't want to. The schematic you posted is just a hint, showing a typical application of the chip, it's not the only way to use it.  

  What would happen if i would not put that capacitor on the input?  


It depends wholly on what you connect to the inputs and outputs. To pass the DC component in the first place you'd have to DC couple AMPOUT too (as well as ATTIN and ATTOUT if you wish to use the attenuator at DC, probably those decoupling pins of the attenuator too). Good luck doing that without messing up all the internal biasing of the amplifier, however. I doubt that it could be done, nor that this is the right tool for the job.  


I agree that the manufacturers are misleading here. It can't really do actual DC under practical terms, but even 1 MHz is functionally DC when compared to 6 GHz.