Question: Why a holomorphic function is infinitely differentiable just because of satisfying the Cauchy Riemann equations, but on the other side, a two variable real function that is twice differentiable is not infinitely differentiable?

I'm asking this for two reasons:

1) $\mathbb{R}^2$ is somehow an analog to the complex plane (they are isomorphic).

2) Holomorphic means that is differentiable in any direction of the complex plane, so why if a function is differentiable in any direction of the real plane then it is not infinitely differentiable?

Answer: 
Even though $\mathbb R^2$ and $\mathbb C$ are isomorphic as real vector spaces, they are very different in some algebraic respects, which crucially influence the notion of differentiability. Recall the key idea that a function is differentiable at a point if it has a best linear approximation (more precisely, a constant plus a linear transformation) near that point. 

In the context of functions $\mathbb R^2\to\mathbb R^2$, "linear transformation" means a transformation that respects addition of vectors and multiplication by real scalars. In other words, the transformation respects the real vector space structure of $\mathbb R^2$. It is well known that such linear transformations are given by $2\times2$ real matrices (once one has chosen a basis for $\mathbb R^2$). 

In the context of functions $\mathbb C\to\mathbb C$, on the other hand, "linear transformation" means a transformation that respects addition of vectors and multiplication by *complex* scalars. In other words, the transformation respects the complex vector space structure of $\mathbb C$. It is well known that such linear transformations are just multiplication by a single complex number. That's much more restrictive than multiplying by an arbitrary $2\times 2$ real matrix.

Specifically, if we use $\{1,i\}$ as our basis for $\mathbb R^2$, then the $2\times 2$ real matrices that correspond to *complex* linear transformations are just those of the form $\pmatrix{a & b\\-b & a}$. Because complex linear transformations are a very special sort of real linear transformations, complex differentiable functions are a very special sort of real differentiable functions. That "specialness" ultimately accounts for all the miraculous consequences of complex differentiability.
This is one of the beautiful features of complex analysis. The answer is that holomorphic functions can be represented by an integral formula (the Cauchy integral formula), whereas most continuously differentiable functions (even smooth ones) on $\mathbb R^2$ cannot.