The grounding wire is connected to prevent to spread the damages and risks when an appliance have a short-circuit.
There will be no problem on how the appliance will operate even if the appliance is not grounded, but for additional safety you can also connect the grounding wire for appliances that are prone to short-circuit in the areas close to water, as washing-machines or refrigerators, it is safer to plug the grounding wire.
※There is no issue plugging more than 1 appliance (ground wire) to a grounding socket.
I.E. you can plug the ground wires from both the refrigerator and microwave to the same socket if the power plugs are close.
Check the instructions below for how to plug the ground wire.
There are types that uses a screw and others that you can just fit the ground wire in the socket.
Check the differences below: