Consider the case when $F=f_{frictional} $. Then
acceleration of the top block is 0, however the bottom block has an
acceleration $a=\frac {F} {M} $. Doesn't it seem unlikely? What am I
missing?
What you are missing is the applied force $F$ cannot equal the friction force. When approaching problems like this it is essential to draw a free body diagram (FBD). See the FBD below. It is based on your subsequent clarification that there is no friction between the bottom block and the surface underneath.
As long as the maximum possible static friction force, $\mu_{s}mg$, between the blocks is not exceeded the two blocks will move together. Ignoring all the vertical forces which sum to zero, there are two forces acting on the top block, the applied external force $F$ and the static friction force $f_f$ that the lower block applies to the top. There is only one external force acting on the bottom block, the friction force applied to it by the top block which is equal and opposite to the friction force applied to the top block by the bottom block.
The acceleration of the top block is
$$a_{m}=\frac {F-f_{f}}{m}$$
The acceleration of the bottom block is
$$a_{M}=\frac {f_f}{M}$$
As long as $F$ doesn't exceed the maximum static friction force between the blocks, the top and bottom blocks move together and therefore
$$a_{m}=a_{M}$$
Equating the two accelerations and doing some math gives us
$$f_{f}=\frac{F}{(1+m/M)}$$
The two forces $F$ and $f_f$ are therefore not equal.
Hope this helps.
