The generating functional is defined as: $$Z[J] = \int \mathcal{D}[\phi] \exp\Big[\frac{i}{\hbar}\int d^4x [\mathcal{L} + J(x)\phi(x)]\Big].$$
I know this object is used as a tool to generate correlation functions by taking functional derivatives, but does it have any interpretation on its own? In this post the answer states that it can be interpreted as the sum of all possible Feynman diagrams. Is this interpretation from Taylor expanding the exponential in $Z[J]$?
The generating functional by definition is an object that encodes the Green's functions in a series as $$Z[J]=\sum_{n=0}^\infty \dfrac{1}{n!}\int dx_1\cdots dx_n J(x_1)\cdots J(x_n) \langle \phi(x_1)\cdots \phi(x_n)\rangle\tag{1}.$$
This is the case because it is defined so that the correlators can be extracted as $$\langle \phi(x_1)\cdots\phi(x_n)\rangle = \dfrac{\delta^n Z[J]}{\delta J(x_1)\cdots \delta J(x_n)}\bigg|_{J=0}\tag{2}.$$
The fact that $Z[J]$ is given by the expression you wrote can then be seem as a consequence of the Dyson-Schwinger equations, for example. See this answer of mine to a question you asked previously for more into this.
So, answering your current question, since each correlator is a sum of Feynman diagrams it is indeed possible to see $Z[J]$ as a sum over all possible Feynman diagrams in view of its definition, equation (1).
Perturbatively, one can formally argue that the partition function/path integral/functional integral/generating functional can be written as $$\begin{align} Z[J]~=~&\int {\cal D}\frac{\phi}{\sqrt{\hbar}}~\exp\left\{ \frac{i}{\hbar}\left(S[\phi] +J_k \phi^k \right)\right\} \cr ~\sim~& \underbrace{\exp\left\{\frac{i}{\hbar} S_{\neq 2}\left[ \frac{\hbar}{i} \frac{\delta}{\delta J}\right] \right\}}_{\text{bag of vertices}} \underbrace{\exp\left\{- \frac{i}{2\hbar} J_k (S_2^{-1})^{k\ell} J_{\ell} \right\}}_{\text{bag of propagators}} , \end{align}$$ cf. my Phys.SE answer here. All possible diagrams are generated by performing the functional $J$-differentiations.
