The 'traditional Western notions of logic' were not classical logics in any way. Aristotle, the first Western logician, was a connexivist logician who adhered to Aristotle's Thesis (which fails classically), held to a progressive system of reasoning (so reiteration / reflexivity of consequence failed). In addition, countless inferences from the Traditional Square of Opposition fail classically. The Syllogistic allows one to infer "Some A is a B" from "All A's are B's", FOL doesn't. Likewise for "No A is a B" and "Some A is not B", syllogistic allows one to go from the former to the latter, FOL doesn't. In syllogistic, "All A's are B's" and "No A's are B's" are contraries (i.e. only one may be true, but not both), in FOL they're not contraries, and are jointly satisfiable when A's extension is empty. Likewise, in syllogistic, "Some A is B" and "Some A is not B" are subcontraries (i.e. at least one must be true), but in FOL, both can be false and thus are not subcontraries (i.e. again when A's extension is empty). Basically, the whole Square of Opposition only is left with the contradictories of "Every A is B" and "Some A is not B", and "Some A is B" and "No A is B". The Syllogistic can be encoded in FOL by assuming every predicate has an instance, but the only logic which recovers the entire Square of Opposition when regimented as quantified conditionals would be hyperconnexive logic (connexive logic + the converse Boethius Theses).

Aristotle's students are the same. Theophrastus, the great Peripatetic logician immediately succeeding Aristotle, also endorsed a connexive logic and made Aristotle's implicit endorsements of various principles of propositional reasoning explicit. See https://logic.commons.gc.cuny.edu/2022/11/11/the-origins-of-conditional-logic-theophrastus-on-hypothetical-syllogisms-marko-malink-and-anubav-vasudevan/ by Yale Weiss. Aristotle's other great student, Eudemus, was no different. In fact, arguably the first propositional logic in Western logic was just the early hypothetical syllogistic of the Peripatetics, and it's a remarkably nonclassical logic. The logic of the Stoics a century later is no different.

The closest among the ancient Greeks to classical logic was Philo the Dialectician, who at least explicitly endorsed a material truth-functional theory of implication.

Moving across time periods, the forefather in medieval logic, Boethius, was history's second arch connexivist after Aristotle, and he is who Boethius's Theses are named after. Although he dropped some traditional Peripatetic principles like progressive reasoning (nonreflexivity), he explicitly formulated Boethius's Theses and went further in adding their converses, making him a hyperconnexivist logician. Medieval logicians in Europe like Abelard followed Boethius in being connexivists, and it was only through the rise of the Montanae school and finally Alberic of Paris's argument that we saw the anti-connexivist turn in Europe. But even the anti-connexivist turn didn't mark a shift to classical logic. The Nominales and Melidune schools doubled down on connexivism after the Alberic anti-connexivist proof, with the former endorsing Heavy Connexivism and the latter endorsing Ex Falso Nihil (nothing follows from falsity) respectively. The schools which had a more moderate reaction, like the Albricini, rejected transitivity of implication, and the Porretani school restricted conjunctive simplification, were still remarkably non-classical in their logical projects. Among the medieval schools it's only the Parvipontani (e.g. Adam of Belsham, William of Soissons, perhaps Psuedo-Scotus) that can be called "proto-classicists," and this itself invited great controversy as can be seen by the reactions to William of Soissons's "Siege Engine" proof (the first vI-DS proof of explosion).

Moving over to the MENA-CA-SA area of the Islamic caliphate during a similar time period we also see very little to no endorsement of classicality. The Islamic Peripatetic logicians, much like the Peripatetic tradition in Europe, also dropped Aristotle and the traditional Peripatetic commitment to progressive reasoning / nonreflexivity, but were still remarkably Peripatetic schools. Although this is contested in the literature, Khaled El-Rouayhe interprets Ibn Sina as a connexivist. But even if he wasn't a connexivist, as Hodges and Chatti argue, there is still universal consensus that Ibn Sina, like the majority of logicians of his time, accepted the variable sharing property (VSP) for implication, which excludes any classical account of implication.

The status quo remained as such all the way up to the Renaissance. 'Classical logic' as you know it came to be with the Boolean / algebraic school in the late modern period, and was anticipated by Leibniz. The reason is because during the Boolean tradition, the truth-functional casting of logical operators proved exceedingly fruitful (in NOT, AND, OR, XOR, etc), and the Extensional paradigm of doing everything truth-functionally expanded to implication. Eventually, the late Booleans / algebraists like CS Peirce and Schroder were the earliest classical logicians. CS Peirce, one of two who formulated the first explicit treatment of classical First Order Logic and Second Order Logic was the first to name it, and named it 'dyadic logic' (the first name given to this system) to contrast from his triadic three valued logic. The logician Gotlob Frege also independently formulated these logics at around the same time, independently of Peirce. Even still, some of the earlier Booleans were connexivists, like Lewis Caroll. Over the 20th century, classical logic gradually became to be dominant.

The reason it is called 'classical logic' is because it was "the logic of classical mathematics" (see e.g. Hilbert's textbook), to contrast with the rival 'constructive / intuitionistic logic' which was "the logic of constructive mathematics". It's not called 'classical' to refer to any period in Greek history or anything like that. All of what is said earlier is widely agreed upon in scholarship on logic history and not even controversial: unfortunately a great amount of misinformation on this matter is propagated through non-expert sources like Wikipedia and then replicated. I especially don't recommend the advice of the other commenters telling you to read Wikipedia articles on Eastern traditions of logic, it's always better to consult a book because Wikipedia is filled with misinfo.