In solution of this problem the acceleration of all the blocks is taken same as $a$ . But won't the acceleration of Block $A$ and $B$ be different and the acceleration of Block $C$ would be the sum of acceleration of $A$ and $B$. Please explain.
Solving this problem requires additional assumptions which are not stated.
For this type of high-school/freshman question you should assume block C is frictionlessly constrained to move only vertically and to remain level by the channel in which it runs.
If it is not constrained to be level, or if it can move from side to side, then it could either tilt (if the strings are attached at different points) or move from side to side. This complicates the analysis, and I haven't attempted to solve for that.
Notice, both the strings of blocks A & B are tied to the (same) block C which moves vertically downward with a certain acceleration. Block C pulls & causes both the blocks A and B to move with the same acceleration. However the tensions in both the strings will be different depending on the masses of blocks and the friction conditions.
Block C can only accelerate in one manner, and because they are tied by strings, Blocks A and B must accelerate at that same rate. Otherwise the strings would either go slack or snap -- in either case the block attached to that string would cease to accelerate because there wouldn't be any force on it.