Today we think of it as inherently Eastern, but it's association with the East is really all via schools of thought that originated in India and then spread further east mainly via Buddhism.
Janism and Buddhism are the oldest still existing belief systems in India that included Reincarnation from the start. Modern Scholars do believe the idea existed among Vedic people already before them but the direct smoking gun proof is lacking. As far as the different theories on when exactly Mahavira aka Vardhamma and Gautama Buddha lived goes, I'm inclined to favor the more recent models which agrees with many other modern historians, so that would be 540-468 BC for Mahavira and 480-400 BC for The Buddha.
Reincarnation existed in Greek thought even at the dawn of Classical Antiquity, the Greek term for it was Metempsychosis. Pythagoras (570-495 BC) was the chief popularizer but at least one Philosopher definitely taught it before him, Pherecydes of Syros who lived 580-520 BC. But the idea could be even older with some speculating both of them got it from the Orphic Mysteries, thing is it's difficult to be certain what any of the old Mystery Cults taught, but the Orphics seem to be the origin of the whole Dionysus being basically a reincarnation of Zagreus thing.
Now this view was never the most popular opinion on the Afterlife in Hellenistic thought, in my opinion Socrates probably did not actually teach it since it doesn't come up in Xenophon's Socratic dialogues, I consider Xenophon far more trustworthy then Plato as a quoter of Socrates. Metempsychosis in Plato's Socrates is I feel the result of Pythagorean influence on Plato during the later decades of his life. It was never Greece's mainstream afterlife view which was the Soul being stuck in Hades forever, nor was it a factor in Stoic or Epicurean thinking. And of course once Christianization happened it was mostly wiped out of Greco-Roman thought, many Pythagorean/Platonist ideas that I consider contrary to Scripture were adopted by the Greek and Latin Fathers, but Metempsychosis was the one idea even the most enthusiastic Platonists knew they couldn't make compatible with The Bible, only a few of the full blown Gnostics even tried.
There is a good deal of evidence that the Ancient Celts/Gauls believed in Reincarnation, Alexander Cornelius Polyhistor, Julius Caesar, and Diodorus Siculus all attest to it, Valerius Maximus and Pomponius Mela also seemingly allude to it. Hippolytus of Rome and Clement of Alexandria speculated on whether or not this implied a connection between them and Pythagoras.
Evidence of it in Ancient Germanic Paganism also exists. Appian in the Second Century said the Teutons believed in Reincarnation, the Poetic Edda also has some allusion to a belief in Reincarnation. This Germanic form of Reincarnation was revived by some Nazis like Heinrich Himmler who believed himself to be a reincarnation of Henry The Fowler, and there's even a quote about Reincarnation attributed to Hitler himself.
The Thracians are also speculated to have had a belief in Reincarnation. If the theory of an Armenian origin for Plato's Myth of Er is correct, then that's another Indo-European people who may have had the concept in their now mostly lost Ancient Pagan beliefs.
There was more contact between Classical Greece and India then people realize thanks to the Silk Road. But even if Reincarnation beliefs in India did develop before or independent of Metempsychosis in Greece, it still originated among the Indo-Iranian speaking peoples, proper ancient Dravidian Folk religion doesn't have it.
When it comes to the Far East I know more about Japan then anywhere else thanks to my Weeb tendencies. Japan has a more fluid approach to Religion then most countries, so today belief in Reincarnation may exist there among people in Japan who wouldn't properly be classified as Buddhists. And for our oldest sources even the Kojiki and the Nihon Shoki weren't written down till after Buddhist presence in Japan was established. But I still feel it's apparent from studying them that Reincarnation was not a part of Japanese thought prior to Buddhist influence.
Taoist texts alluding to belief in something like Reincarnation don't show up till the Han period (200 BC to 200 AD), and even then it's more like Avatars of Vishnu in Hinduism not Reincarnation for Normal non Divine people. Taosim in general even today doesn't generally teach reincarnation, rather their view on The After Life is the basis for the After Life presented in Star Wars (as well as a lot about The Force itself, see Rick Worley's video on the Whils.). Indeed if I were to compare Taoism to a school of Classical Greek Philosophy it would be the Stoics rather then Pythagoras or Plato.
Now I know attributing to "The West" something popularly thought of as Eastern is often something White Supremacists do. But I do not believe in Reincarnation or consider it's influence on the world a good thing. I have enjoyed some fiction that involves Reincarnation in it's premise, like some of that Anime I'm into and The Legend of Zelda. But I think seriously believing in it in real life can prove to be quite dangerous.
