Let me try to say something meaningful about the following:

"(1) Can someone point me to similar definition in case of sheaf cohomology on a scheme. (2) What is the necessity to go to so called complex of abelian sheaves? What am I missing here?"

Let me first say something general about (2). Take your example of the algebraic de Rham complex. As Keerthi mentions in the comments, a complex is richer than its (co)homology, but not only this, a complex is richer than an abelian sheaf all on its own. This is because there is an embedding of abelian sheaves into the category of chain complexes of abelian sheaves. Not only this, but sometimes there are invariants of a scheme which cannot be encoded in a single sheaf. An example of this is the algebraic de Rham complex. It is a special cochain complex of coherent sheaves on $X$, in that it has "multiplicative structure", meaning that there are maps

$$\Omega_{X}^{p} \otimes_{\mathscr{O}_{X}} \Omega_{X}^{q} \rightarrow \Omega_{X}^{p+q}$$

compatible with the cochain complex structure of $\Omega_{X}^{\ast}$ (this is wedging forms together).

Now, suppose that we are given a bounded below cochain complex of abelian sheaves, $\mathscr{F}^{\ast}$ (more generally, just a bounded below cochain complex of objects in an abelian category). The sheaf hypercohomology of the complex $\mathscr{F}^{\ast}$ is to $\mathscr{F}^{\ast}$, is what the sheaf cohomology of some abelian sheaf $\mathscr{G}$ is to $\mathscr{G}$. What I mean is that the sheaf hypercohomology of a complex of sheaves is obtained by taking some appropriate resolution of $\mathscr{F}^{\ast}$, applying the global sections functor to it, taking the totalization of this double complex, and then taking the cohomology of the totalized complex. This is roughly the same procedure as resolving a sheaf via an injective/acyclic resolution, applying the global sections functor, and then taking the cohomology. Let me spell this procedure out in greater detail:

1. Take a Cartan-Eilenberg resolution $I^{\ast,\ast}$ of the chain complex $\mathscr{F}^{\ast}$ too obtain a first-quadrant double complex (see Chapter 5 of Weibel's "Introduction to Homological algebra").

2. Apply $\Gamma(X,-)$ to $I^{\ast,\ast}$ to obtain a double complex in $Ab$.

3. Then compute $H^{n}\left( Tot^{\ast}(I^{\ast,\ast}(X)) \right)$, where $Tot^{n}(I^{\ast,\ast}(X))= \prod_{p+q=n} I^{p,q}(X)$. This is just $\mathbb{H}^{n}(X,\mathscr{F}^{\ast})$.

More generally, we can compute the hyper-derived functor of a right/left exact functor. Ultimately, the derived category encodes all this information, but it takes a some work to see why.

Let me try to say something meaningful about the following:

"(1) Can someone point me to similar definition in case of sheaf cohomology on a scheme. (2) What is the necessity to go to so called complex of abelian sheaves? What am I missing here?"

Let me first say something general about (2). Take your example of the algebraic de Rham complex. As Keerthi mentions in the comments, a complex is richer than its (co)homology, but not only this, a complex is richer than an abelian sheaf all on its own. This is because there is an embedding of abelian sheaves into the category of chain complexes of abelian sheaves. Furthermore, sometimes there are invariants of a scheme which cannot be encoded in a single sheaf. An example of this is the algebraic de Rham complex. It is a special cochain complex of coherent sheaves on $X$, in that it has "multiplicative structure", meaning that there are maps

$$\Omega_{X}^{p} \otimes_{\mathscr{O}_{X}} \Omega_{X}^{q} \rightarrow \Omega_{X}^{p+q}$$

compatible with the cochain complex structure of $\Omega_{X}^{\ast}$ (this is wedging forms together).

Now, suppose that we are given a bounded below cochain complex of abelian sheaves, $\mathscr{F}^{\ast}$ (more generally, just a bounded below cochain complex of objects in an abelian category). The sheaf hypercohomology of the complex $\mathscr{F}^{\ast}$ is to $\mathscr{F}^{\ast}$, is what the sheaf cohomology of some abelian sheaf $\mathscr{G}$ is to $\mathscr{G}$. What I mean is that the sheaf hypercohomology of a complex of sheaves is obtained by taking some appropriate resolution of $\mathscr{F}^{\ast}$, applying the global sections functor to it, taking the totalization of this double complex, and then taking the cohomology of the totalized complex. This is roughly the same procedure as resolving a sheaf via an injective/acyclic resolution, applying the global sections functor, and then taking the cohomology. Let me spell this procedure out in greater detail:

1. Take a Cartan-Eilenberg resolution $I^{\ast,\ast}$ of the chain complex $\mathscr{F}^{\ast}$ too obtain a first-quadrant double complex (see Chapter 5 of Weibel's "Introduction to Homological algebra").

2. Apply $\Gamma(X,-)$ to $I^{\ast,\ast}$ to obtain a double complex in $Ab$.

3. Then compute $H^{n}\left( Tot^{\ast}(I^{\ast,\ast}(X)) \right)$, where $Tot^{n}(I^{\ast,\ast}(X))= \prod_{p+q=n} I^{p,q}(X)$. This is just $\mathbb{H}^{n}(X,\mathscr{F}^{\ast})$.

More generally, we can compute the hyper-derived functor of a right/left exact functor. Ultimately, the derived category encodes all this information, but it takes a some work to see why.