Why pairing based crypto is suitable for some particular cryptographic primitives? Why pairing based crypto is being widely used in some special crypto primitives as ID based crypto and variations of standard signatures? I mean taking as deep as possible what makes it suitable for that schemes while other schemes do not feet?
Crypto based on cyclic groups is (at a very high level) about "hiding" things "in the exponent" and then manipulating those values as they live in the exponent. As an example, in a cyclic group $\langle g\rangle$, you can "hide" a random value $x$ as $g^x$.

Without a bilinear pairing, all you can really do "in the exponent" are linear/affine (degree-1) combinations of these hidden values. That is, given $g^{x_1}, \ldots, g^{x_n}$, you can obtain $g^{a_0 + a_1 x_1 + \cdots + a_n x_n}$ for known coefficients $a_i$.

With a bilinear pairing, you can do degree-2 combinations. This is huge, it allows you to "multiply hidden values together in the exponent". This extra expressivity is what lends itself to a wider variety of cryptographic primitives, like identity-based and functional encryption, etc.

There are other tools like lattices that also have a lot of algebraic structure. The capabilities of lattices are somewhat incomparable to those of groups with bilinear pairings. But more of these "functional encryption" applications are now being achieved using lattices as well. So none of these applications are probably unique to bilinear pairings, it is a bit of a historical accident that we have seen an explosion in techniques and applications of bilinear pairings.