This got longer than I thought; skip to the end if you just want a couple possibly-useful references!
There is, unsurprisingly, extensive work on algebraic and topological aspects/interpretations of first-order logic. But you are correct that it is much murkier than the propositional setting.
The most obvious issue is that semantics and syntax are much further separated in the case of $\mathsf{FOL}$. A "model" for propositional logic in a fixed language (= set of atomic propositions) $P=\{p_i:i\in I\}$ is just a valuation map $\nu:P\rightarrow\{\top,\perp\}$, and there is no meaningful difference between models and complete (= maximal consistent) theories. The situation for first-order logic is completely different: even if you restrict to "nice" structures (e.g. countable ones), there are non-isomorphic structures with identical theories. In fact, there is no canonical way to associate a single model up to isomorphism with a complete theory, in the sense that $\mathsf{ZF}$ alone cannot prove that for each (first-order) language $\Sigma$ there is a function sending each complete $\Sigma$-theory to a model of that theory.
The situation is even worse than this indicates, though. The above might suggest that we could "fix" things by (assuming Choice and) using a stronger logic than $\mathsf{FOL}$. But there really is a fundamental jump in complexity of the semantics involved which can't be gotten around. Forgetting logics and theories entirely for a second, each $P$-valuation is determined entirely by its restrictions to the individual $p_i\in P$, and the space of $P$-valuations is the $I$-fold product of these atomic valuations. By contrast, given a first-order language $\Sigma=\{R_j:j\in J\}$, there is no good way to combine a family $\{\mathcal{M}_j:j\in J\}$ with each $\mathcal{M}_j$ an $\{R_j\}$-structure into a single $\Sigma$-structure, appropriate forgetful functors notwithstanding. In the first-order setting, the different parts of the language can interact meaningfully with each other, and that's not true of the propositional setting. This is what leads the obvious attempt to topologically prove the compactness of first-order logic to fail.
The topological analyses of first-order logic and general predicate logics are significantly more complicated than their propositional counterparts; see e.g. type spaces (which are Stone spaces!). For that matter, so are the algebraic analyses (cf. cylindrical algebras or polyadic algebras). So we should expect any duality theorems around first-order logic to be similarly complicated. I'm not an expert here, but at a glance Makkai-style conceptual completeness theorems and papers like van Gool/Marques' On duality and model theory for polyadic spaces seem relevant. In particular, quoting from Makkai's paper: "Strong conceptual completeness is most familiar in (classical) propositional logic, where it takes the form of the Stone duality theorem. "
