Relation between OSAs and grammars
I find the answer to my own question in [2]:

Let $G = \langle N, \Sigma, P \rangle$ be a context-free grammar. We can make $G$ into a many-sorted algebra by taking $N$ as the set of sorts and

$$
G_{w, n} = \{\, p \in P \,\mid\, p = (n, w') \land \operatorname{nt}(w') = w \,\}
$$
as an $N^* \times N$-sorted family ($\operatorname{nt}(\alpha)$ returns the ordered sequence of non-terminal symbols in $\alpha \in (N \cup \Sigma)^*$).

The initial $G$-algebra contains the parse trees for all the derivations in the grammar.

Conversely, if $\Sigma$ is an $S$-sorted operator domain, the initial algebra obtained under the previous construction with the grammar

$$
\hat{\Sigma} = \langle S, \Sigma \cup \{(,)\}, P \rangle\quad P=\{\, (s, \sigma(s_1\cdots s_n)) \,\mid\, \sigma \in\Sigma_{s_1\cdots s_n,s} \,\}
$$

is isomorphic to the initial $\Sigma$-algebra.



[2] Goguen, Joseph A., et al. "Initial algebra semantics and continuous algebras." Journal of the ACM (JACM) 24.1 (1977): 68-95.