Is the force exerted by the liquid, in which the object is floating, same as the buoyant force exerted by the liquid? I am confused since the object is assumed to float with some part of it above the liquid surface.
Yes the force exerted by the liquid on a floating object is the same as its buoyant force.
Part of an object floats above the liquid surface simply because the density of the object (Mass/unit volume) is less than the liquid.
If the density of the object is greater than the liquid, it will sink on its own until, or unless, its density becomes equals to that of the liquid. That may happen at sufficiently large depths as density of a liquid generally increases with depth due to increased pressure, at which depth it will stop sinking.
If the density of the object equals the liquid, the object can be positioned at various submersed levels and it should remain stationary, as long as the densities remain the same.
Hope this helps.