Two vessels have the same base area but different shapes. The first vessel takes twice the volume of water that the second vessel requires to fill upto a particular common height. Is the force exerted by the water on the base of the vessel the same in the two cases? If so, why do the vessels filled with water to that same height give different readings on a weighing scale?

Two vessels having the same base area have equal force and equal pressure acting on their common base area.
The force exerted on the sides of the vessels has non-zero vertical components because the shapes of the two vessels are different. On adding these vertical components, the total force on one vessel comes out to be greater than that on the other vessel.
Hence, when these vessels are filled with water to the same height, they give different readings on a weighing scale.