I was wondering in light of the historical developments of logic since ancient Greeks and well into the nineteenth and twentieth centuries: What kind of a philosophy assumes only one Logic, and what philosophical premises are given up once permitting all kinds of logic? (e.g. classical logic, intuitionist logic, many-valued logic, and so on.) Is it the case that once assuming only one Logic we actually ascribe Logic a metaphysical status in the form of say "Laws of thought", and once allowing all sorts of Logic we rather de-ontologize Logic and turn it into mere instrument?
It is more complicated than a merely instrumental/metaphysical division. The "logics as mere instruments" were not meaningful until well into the 20th century, after the general disillusionment in foundationalist epistemologies, and the subsequent pluralism of formal systems and interpretations. Originally, Logic meant the study of Logos, reason, its laws, how it functions, etc., the meaning closer to what we now call epistemology. This is how Plato, Spinoza, Leibniz, Kant, Hegel, Peirce, Husserl, etc. understood Logic, and it is closer to what is now called epistemology, although "logic" is still used in the old sense by some continental philosophers (Adorno, Deleuze).
When Boole, De Morgan and Peirce produced first logical algebras they were just formalizing some aspects of that grand Logic, not shattering it into multiple mini-logics. Indeed, the logicism of Frege and Russell, the idea of a universal logical shell, was a derivative of this grand meaning, only they dreamed, along with Leibniz, that human reason, or the scientific side of it at least, could actually be formalized wholesale. In Carnap's late version of logicism, following the foundational crises in both physics and mathematics, this universal shell was already shattered into multiple "linguistic frameworks", but still with the analytic/synthetic distinction left in place.
Formal logic, like the syllogistic, was only the most elementary part of Logic, although Aristotle and scholastics emphasized it enough to put them closer to current usage. Even so, Aristotle's discussions of anchinoia, guessing the middle premise in the so-called synthetic syllogism, fall under what is now called abductive inference and is not part of logic in the narrow sense (although some analytic philosophers follow Peirce in talking about "logic of discovery" in a non-Popperian sense). For a description of the current positions see informative discussion in D’Agostini's From a Continental Point of View: The Role of Logic in the Analytic-Continental Divide:
"Analytic philosophers usually consider formal logic indispensable in philosophical training, while continental philosophers often despise logic and consider it a barren and useless subject... The disagreement regards not only the use of logic, but its nature and definition as well. Possibly, there is no common idea of logic clearly shared by the opponents...
The most relevant disagreement probably concerns the first [narrow] meaning: while analytic philosophers have fundamentally accepted the new formal science created by mathematicians, continental philosophers generally hold that philosophy should use a specific logic (dialectical, transcendental, hermeneutical, etc.), or even be free of any logical commitment. In contrast, analytic and continental philosophers might have a useful dialogue over the second [grand] meaning i.e., over logic as pure theory. Indeed, it is difficult to deny that they have both been in some way concerned with the misfortunes of theoretical reason after Kant".
From the Stanford Encyclopedia of Philosophy (the link provided by @Conifold in his comment to his answer):
Logical pluralism is the thesis that there is more than one correct logic. The main opposing view, logical monism, is the thesis that there is only one.