Expanding a bit more on my answer to the linked question, propositional logic is indeed a bit silly insofar as we're interested in logic from the perspective of classifying or analyzing mathematical structures. This isn't to say that it's wholly uninteresting, but certainly it enjoys much narrower interest than, say, first-order logic.
A propositional language is just a set of propositional atoms $P=\{p_i:i\in I\}$; a "structure" in this language is just a valuation, that is, a map $$v:Sent_P\rightarrow \{\top,\perp\}$$ (where "$Sent_p$" is the set of sentences which can be formed from the atoms in $P$) satisfying some basic properties such as $v(a\wedge b)=\top\iff v(a)=\top$ and $v(b)=\top$. The satisfaction relation is then just $$v\models\varphi\iff v(\varphi)=\top.$$
Now already a valuation doesn't seem particularly interesting, but it gets worse:
A valuation $v$ is determined entirely by its behavior on $P$, that is, by $\{i: v(p_i)=\top\}$.
Conversely, any map $v_0: P\rightarrow\{\top,\perp\}$ extends to a
unique valuation: for every such map $v_0$ there is exactly one valuation $v$ with $v\upharpoonright P=v_0$.
And a map $v_0:P\rightarrow\{\top,\perp\}$ can be equated with the set $v_0^{-1}(\top)$. So we can simplify things even further: our "semantic universe" is just $\mathcal{P}(P)$, the set of subsets of $P$.
By contrast, a structure in the sense of first-order logic is a much more interesting object: it's a set equipped with some "named" constants, functions, and relations (according to the language involved). Many mathematical structures are in fact structures in this precise sense, such as groups, rings, fields, etc. For example, the class of $\{*\}$-structures (for $*$ a binary function symbol) contains (in a precise sense) the class of all groups! So the semantics of first-order logic is quite broad and interesting. In particular, we frequently care about the first-order theories of specific first-order structures (e.g. the ring of reals is fundamentally less complicated than the ring of integers per Tarski and Godel) but we almost never care about the propositional theories of propositional "structures" (outside of a complexity-theory context, from which perspective propositional logic is indeed intersting).
