Below, all structures are countable and in finite languages. This question is motivated by the following: "To what extent is self-reference possible when nothing like the diagonal lemma holds?"
Somewhat-informally (let me know if I should add more details), say that:
Given two structures $\mathcal{B},\mathcal{A}$ in possibly-different languages, a system of interpretations of $\mathcal{B}$ into $\mathcal{A}$ is a pair $(\Phi(\overline{x};\overline{y}),\upsilon(\overline{y}))$ such that
$\upsilon$ is an $\mathsf{FOL}$-formula with $\upsilon^\mathcal{A}\not=\emptyset$ and
$\Phi$ is a tuple of $\mathsf{FOL}$-formulas such that, whenever $\overline{b}\in\upsilon^\mathcal{A}$, we have $\Phi(\overline{x};\overline{b})$ is an interpretation of $\mathcal{B}$ into $\mathcal{A}$ in the usual sense (in particular, parameter-free except for $\overline{b}$).
A syntax structure over $\mathcal{A}$ is an expansion (possibly including additional sorts) $\mathcal{X}$ of the pure set $S$ of sentences in the language of $\mathcal{A}$, such that every automorphism of $\mathcal{X}$ fixes $S$ pointwise.
Now given a structure $\mathcal{A}$ in a language $\Sigma$, let the self-referentiality spectrum of $\mathcal{A}$ be the set $\mathsf{SR}(\mathcal{A})$ consisting of all $\Sigma$-formulas $\theta$ satisfying the following property:
There is a syntax structure $\mathcal{X}$ over $\mathcal{A}$ such that $(i)$ $\mathcal{X}$ is interpretable-without-parameters in $\mathcal{A}$ and $(ii)$ whenever $(\Phi,\upsilon)$ is a system of interpretations of $\mathcal{X}$ into $\mathcal{A}$, there is a function $F$ which is parameter-freely-definable over $\mathcal{A}$ such that whenever $\overline{b}\in\upsilon^\mathcal{A}$ we have that $F(\overline{b})$ is the $\Phi(-;\overline{b})$-code for a sentence $\sigma$ with $$\mathcal{A}\models \sigma\leftrightarrow \theta(F(\overline{b})).$$
(Of course $\theta(F(\overline{b}))$ is a sentence with parameters from $\mathcal{A}$, but that's fine.) Note that as the tuple $\overline{b}$ varies, the "meaning" of $\theta$ varies as well; in particular, $F$ almost certainly can't be constant. For example, $\mathsf{SR}(\mathbb{N};+,\times)$ is the set of all formulas in the language of arithmetic, while the much weaker structure $(\mathbb{N};+)$ has at least $\top$ and $\perp$ in its self-reference spectrum.
I suspect that $\mathsf{SR}(\mathbb{N};+)$ is "close to trivial," and in particular that at least its unary elements should be easy to describe:
Question: Is it the case that every unary formula in $\mathsf{SR}(\mathbb{N};+)$ defines a finite-or-cofinite subset of $\mathbb{N}$?
