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 up to 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?

(A) Yes,the force exerted by the water on the base is the same in both cases.
The pressure at the base of a vessel depends only on the height of the liquid column $(P = h\rho g)$. Since both vessels have the same base area $(A)$ and are filled to the same height $(h)$,the force exerted on the base $(F = P \times A = h\rho gA)$ is identical for both.
The reason they give different readings on a weighing scale is due to the force exerted by the walls of the vessel on the water. Because the shapes are different,the walls exert vertical components of force on the water. In one vessel,the walls may push the water downwards,while in the other,they may push it upwards. These vertical forces are transmitted to the weighing scale,resulting in different total weights being measured.