1. Here is what Tate says in his account of the General Reciprocity Law in the AMS volume on Hilbert's problems :
With this work of Takagi the theory of abelian extensions --- "class field theory" --- seemed in some sense complete, yet there was still no general reciprocity law. It remained for Artin to crown the edifice with such a theorem. He conjectured in 1923 and proved in 1927 that there is a natural isomorphism $$ > C_K/N_{L|K}C_L\buildrel\sim\over\to\operatorname{Gal}(L|K) > $$ which is characterised by the fact that...
And a little later :
How did Artin guess his reciprocity law ? He was not looking for it, not trying to solve a Hilbert problem. Neither was he, as would seem so natural to us today, seeking a canonical isomorphism, to make Takagi's theory more functorial. He was led to the law by trying to show...
Read him.
2. Here is a toy example --- not unrelated to class field theory --- of how a bijection can be more natural than others. Let $p$ be a prime number and let $K$ be finite extension of $\mathbb{Q}_p$ containing a primitive $p$-th root of $1$. There are only finitely many degree-$p$ cyclic extensions $L|K$, and there are only finitely many vectorial lines in the $\mathbb{F}_p$-space $K^\times/K^{\times p}$. In fact the two sets have the same number of elements, but the only natural bijection is $$ L\mapsto\operatorname{Ker}(K^\times/K^{\times p}\to L^\times/L^{\times p}), $$ of which the reciprocal bijections can be written $D\mapsto K(\root p\of D)$.
It follows that the number of degree-$p$ cyclic extensions $L|K$ is the same as the number of hyperplanes in $K^\times/K^{\times p}$. But is there a natural bijection between these two sets ? You will agree that $L\mapsto N_{L|K}(L^\times)/K^{\times p}$ is as natural a bijection as there can be.