To talk about "arbitrary logics" we need some kind of formalism describing what a logic is in the abstract. A very general definition would be
An abstract logic is a pair of sets $I$ and $S,$ and a relation $\models$ between $I$ and $S.$
Here, the elements of $I$ are called interpretations and elements of $S$ are called sentences and the relation $\models$ is called satisfaction. Then a sentence $\phi\in S$ is called satisfiable if there is an interpretation $\mathcal M\in I$ such that $\mathcal M\models \phi.$
So at this very high level of abstraction, we define our way out of the problem. So your question can be refined to asking if this formalism is applicable to more exotic logics than just classical propositional logic. Yes.
- In first order classical logic the interpretations are structures and satisfaction is defined via Tarski's inductive definition. (And this generalizes to higher order and infinitary classical logics.)
- In modal logic and intuitionstic logic, interpretations can be taken as Kripke frames and satisfaction of a sentence generally means that it holds at every world of the frame (where we then define "holds at a world" inductively on the structure of the formula).
- You allude to many-valued propositional logics. I should emphasize that many-valuedness is not by any means the only generalization (and the nonclassical logics in the above bullet are more important in my opinion$^*$). But yes, in many-valued systems it is usually the case that one of the values intuitively represents "true" and then satisfaction in an interpretation just means that it takes that truth value. Generally, logics where the interpretations assign formulas truth values that are linearly ordered between true and false are called fuzzy logics. Another example mentioned by Alessandro in the comments, is Boolean-valued models where the truth values come from some fixed Boolean algebra, so are only partially ordered, although there is still a top value representing true.
$^*$Actually, intuitionistic logic can be considered to be a many-valued logic since there is a fixed Heyting algebra such that a sentence is intuitionistically valid if and only if any assignment of propositional variables into the algebra results in the sentence receiving the top value.