I think there is a confusion of terminology here.
The multiplicity one theorem that Shalika proved is a global result: it tells us that if two irreducible subspaces of $L^2(\mathrm{GL}_n(\mathbb{Q})\backslash\mathrm{GL}_n(\mathbb{A}))$ are isomorphic (as automorphic representations), then they are equal (as subspaces). In other words, for a given set of local representations $\pi_v$ ($v$ a place of $\mathbb{Q}$), there is at most one irreducible subspace $\pi\subset L^2(\mathrm{GL}_n(\mathbb{Q})\backslash\mathrm{GL}_n(\mathbb{A}))$ whose local factor at each place $v$ of $\mathbb{Q}$ equals $\pi_v$. In practical terms, the result tells us that the complete $L$-function $\Lambda(s,\pi)$ determines $\pi$ as a subspace of $L^2(\mathrm{GL}_n(\mathbb{Q})\backslash\mathrm{GL}_n(\mathbb{A}))$.
The multiplicity one theorem that Goldfeld talks about is a local result: it tells us, in a concrete form, that $\pi_\infty$ is determined by the action of $Z(U(\mathfrak{gl_n(\mathbb{R})}))$ on the vectors in $\pi$. In other words, what they prove is, roughly, multiplicity one at the place $\infty$, while Shalika proves this for all places together. In practical terms, the result tells us that the gamma factors in the complete $L$-function $\Lambda(s,\pi)$ determine $\pi_\infty$.
(I don't know Shalika's proof, but an obvious approach is via the Whitakker model, which harmonizes with the stament in Goldfeld's book. Also, I don't think that Goldfeld claims any originality here; he just gives an elementary treatment of a useful fact he might need later.)
Note that $\pi_\infty$ does not determine $\pi$ uniquely, not even for $n=2$ and $\Gamma_0(N)$. This is because a Laplacian eigenvalue can have multiplicity among the Maass newforms of level $N$ on the upper half-plane. On the other hand, for $N=1$ we do expect that $\pi_\infty$ determines $\pi$ uniquely, but this is a very famous unsolved problem that seems to be out of reach at the moment (it is usually compared to the simplicity of zeros of $\zeta(s)$).
