The Academy, was literally invented by Plato..! I like Vervaeke's point that Plato took the subversive openess to questioning that got Socrates killed, and mixed it with the Pythagorean math-cult, to create academia - creating a school that could reach the high and low-born, to be financially sustainable, but also a place for anyone that could argue really well.
Philosophy, mysticism, religion, who's boxes are these? Why should people who didn't make those boxes, be expected to sort their traditions neatly into them? Confucianism seriously stretches our definition of 'religion'. Hindu & Buddhist thought have deeply philosophical traditions and movements, and almost purely 'mystical' ones, but because they place primacy on subjective experience, mind, over world, these are not incompatible there. Key insights for Greek mathematics, a derivation of pi, geometry, zero & infinity, are thought to have come directly from Indian religious thought.
I would argue that philosophy, as distinct from general wisdom traditions, schools of logic, ethical systems, schools of jurisprudence, etc etc was made distinct specifically by Socrates, and his 'martyrdom' to it. Plato found roots, aspects, in the pre-Socratics, and their proposals of universal-substances is a kind of physico-materialism precursor, separating minds and intentionality from understanding the natural world. But philosophy depends crucially on Socratic dialogue, as opposed to other modes of engagement pursuasion or practice, like rhetoric, or initiations (eg mystery cults, tantra, Tibetan empowerments), or meditation & chanting. Joint commitment to the pursuit of truth, and wisdom. Without making that explicit, I say you don't have philosophy. It is not that Western thought is entirely scientific and physico-materialist, it's that if you commit to free and open debate, the special-pleading, magic books and miracles, aren't pursuasive. A religious person is an atheist to the god/s of every other tradition.
Does that mean it didn't exist before Socrates? I would describe discourses that move toward wisdom through unscripted question and answer as philosophical, so I'd include Confucius and Buddha, as philosophers. But they were other things too, on spirituality, judicial theory etc. Socrates we remember as a philosopher, because he made doing what he did, be being 'a philosopher'.
Why is it called Western? As part of conscious efforts to create a sense of cultural-supremacy to match the political one, of a resurgent backwater that had been largely lost to religious fanaticism for a millenium and a half: Europe. Fortunately, the Islamic world which stretched from India to Morocco had preserved the texts of Greek thought beyond the few fragments that had survived in Europe, and provided critical commentaries like Avicenna's (Ibn Sina) and Averroes (Ibn Rushid), and invented algebra on the side (al jabr, 'reunion of broken parts'). St Augustine, one of the two most important philosophers to Christianity, was from North Africa (from Alexandria, where arguably the library was invented).
For more context-setting on the geographical and political differences that led to different traditions:
Western Empiricism in Eastern Philosophy?