For local field, the reciprocity map establishes almost an isomorphism from the multiplicative group to the Abelian Absolute Galois group. (In global case the relationship is almost as nice). It is tempted to think that there can be no such nice accident. 


Do we know any explanation which suggest that there "should be" a relationship between the multiplicative group and the Galois group?


Actually, my current belief is that the reciprocity map is half accidental. I think that there is a natural extension where we can define a natural action of the multiplicative group. In the local case this is the Lubin-Tate extension (a generalization of cyclotomic extension). The fact that this Lubin-Tate extension is the Abelian Absolute Galois group is an accident. 


Do we know something that might support/ reject this view?


Please feel free to edit the question into a form that you think might be better.



How natural is the reciprocity map?
The reciprocity map is completely natural (in the technical sense of category theory). For example, if $K$ and $L$ are two local fields,
and $\sigma:K \rightarrow L$ is an isomorphism, then $\sigma$ induces an isomorphism of 
multiplicative groups $K^{\times} \rightarrow L^{\times}$ and also of abelian absolute Galois groups $G\\_K^{ab} \rightarrow G\\_L^{ab}$. The reciprocity laws for $K$ and $L$ are then compatible with these two isomorphisms induced by $\sigma$. 


On the other hand the factorization $K^{\times} = {\mathbb Z} \times {\mathcal O}\\_K^{\times}$ is not canonical (it depends on a choice of uniformizer), and the identification of
${\mathcal O}\\_K^{\times}$ with the Galois group of a maximal totally ramified abelian extension of $K$ also depends on a choice of uniformizer (which goes into the construction of the Lubin--Tate formal group, and hence into the construction of the totally ramified extension; different choices of uniformizer will give different formal groups, and different
extensions). 


As others pointed out, the local reciprocity map is also a logical consequence of the global Artin map and global Artin reciprocity law (which makes no reference to local multiplicative groups, but simply to the association $\mathfrak p \mapsto Frob\\_{\mathfrak p}$ of Frobenius elements to unramified prime ideals; see the beginning of Tate's article in Cassels--Frohlich for a nice explanation of this). Thus it is natural in a more colloquial sense of the word as well.


Indeed, the idelic formulation of the glocal reciprocity map and the formulation of the local reciprocity map in terms of multiplicative groups are not accidental or ad hoc inventions; they were forced on number theorists as a result of making deep investigations into the nature of global class field theory.


