But that game is full of utter nonsense.
As for 'Earth Army', as adamant I am that there aren't or shouldn't be Starfleet Marines, I actually think there could be various planetary armies and militias. These however would be more like national guard type organisations, who help in cases of various natural disasters or defend the planet in the unlikely case of an invasion. In some places they might be the same thing as the local police or fire and rescue service. They are not deployed outside their homeplanets and would be run by the member governments, not the Federation. There is no 'Federation Army.'