AMG is a part of mercedes benz now. AMG is the modified version you can say. Initially AMG cars were all body kitted and tuned but now you can order mercs with AMG styling only, that means any model of merc can be ordered with AMG body styling which includes bumpers, side skirts and few other changes in suspension along with wheels. Engine modifications are also there, its all what you order for.
SL55, E55, C55, CLS55, S55 and models like this are pure AMGs with full tuned engines.
Just the badge does not mean anything, Its AED 15 for one badge in dubai (i.e Rs. 250 or so) but fake.
Its like BMW has their M versions but you can also order any BMW with M styling.
Hope its clear.