Step-by-step explanation:
Bear in mind that the 13A fuse in the plug is not related to the 2A that your appliance draws. By the time the appliance causes the fuse to blow it is too late to protect the appliance.
The fuse rating is based upon the cable size between the plug and the point of failure. If the flex is sized to handle the full 3kW load, then a 13a fuse is reasonable. But if it is a lamp with skinny flex that is only intended to carry the 2A of the load, then protecting it with a 13A fuse risks further damage and possible ignition of the cord, and consequential building damage.
If you are aware of the core size of the cable, and have access to a set of copper tables, and know what the upper design limit is on the insulation temperature, then it is reasonable to make a judgement like you suggest.
If not, the easy decision is to rate the plughead fuse to suit the rating plate on the device. You know that the cable is sized to suit at least that load, so it is a sensible way to chose the fuse.