Let $S$ and $T$ be sets with the same algebraic structure. Let $\Phi:S^n\to T$ such that for any $i\leq n$, and $s_1,\ldots,s_n\in S$, the aplication $s\in S\mapsto \Phi[s_1,\ldots,s_{i-1},s,s_{i+1},\ldots,s_n]\in T$ is a morphism (i.e. respect the algebraic structure of $S$ and $T$).
If $S$ and $T$ are vector space, we say that $\Phi$ is multilinear. In the general case do we also say multilinear? does it exist a distinct general denomination?
The term "multilinear" is appropriate everywhere "linear" is, which I would say is in the setting of modules. The term is certainly standard for modules over commutative rings, e.g. when considering tensors on smooth manifolds.
In more general settings "linear" is replaced with "homomorphic", so we might expect "multilinear" to be replaced with "multihomomorphic". Indeed it seems from a web search "bihomomorphism" is common terminology for the case $n=2$ in semigroups, so the "linear" $\to$ "homomorphic" replacement scheme seems good at first. However, searching for "multihomomorphism" shows that the majority of uses are actually describing a multivalued homomorphism of groups or graphs. Thus using "multihomomorphic" seems potentially confusing, despite its naturality.
I'm not an algebraist so all of the above is just based on my thoughts and what Google turned up.