Last year, in a talk of Michel Waldschmidt's, I remember hearing a statement along the lines of the title of this question, that is, "The Galois group of $\pi$ is $\mathbb{Z}$.". In what sense/framework is this true? What was meant exactly  and can this notion be made precise?

I gather that the idea behind $\mathrm{Gal}(\pi)=\mathbb{Z}\backslash\{0\}$ (not $\mathbb{Z}$, $0$ is not a conjugate of $\pi$!) comes from Euler's formula: $$\prod_{n\in \mathbb{Z\backslash\{0\}}}\bigg(1\frac{x}{n\pi}\bigg)=\frac{\mathrm{sin}(x)}{x} \in\mathbb{Q}\{x\} $$ which can make you think of those $n\pi$ as the conjugates of $\pi$. But in order for the Galois groups to act transitively on the conjugates you need all the nonzero rationals, so that $\mathrm{Gal}(\pi)=\mathbb{Q^{\times}}$. For some actual evidence that $\mathrm{Gal}(\pi)=\mathbb{Q^{\times}}$ is the right answer (this is, consistent with the conjectural picture of periods and motives), see sections 3 and 5 of Galois theory, motives and transcendental numbers by Yves Andre, alredy mentioned in the comments. 

