The reason that Europe is thought of as being in the West is because it coincides with the Western Roman Empire
It's not that Europe would be thought of as being in the West nowadays. It's that anything west of the river Oder is thought of being in the West, for very concrete reasons less than a century old.
Why, following the days of the two Romes - that is, in the early days of the third, in Moscow - there was no East vs. West. Muscovy wasn't an enemy one full compass direction wide: there was just Europe, and then the rest of the world, which didn't lie in any particular direction (except of course "India", which was a direction that covered everything that wasn't visible from mainland Europe or the shores of the Mediterranean).
The division of the Trek universe into Alpha and Beta wouldn't be of ancient origin, either - certainly not if the dividing line runs through the upstart Sol system that played no role in galactic politics until about 100-200 years before the events. Indeed, none of the current major players is known to have thought of the galaxy in terms of Quadrants before Earth started making noise...