I'm sure there's a story to be told about why the tradition persisted for so long, but the origin of the custom lies in the history of the names.

The idea of a division between Europe and Asia goes back to the 6th century BCE Greek ethnographer Hecataeus of Miletus. In the Greek world at the time, the important division was between the lands on either side of the Aegean Sea. And even there, both Asia and Eurōpē originally referred to smaller areas -- Asia was originally a part of central western Turkey around the Küçük Menderes river, and Eurōpē was probably originally in the far north of Greece.

And then the Greeks started establishing colonies all over the place. Hecataeus, who was describing the entire world as known to him, found himself in the position of choosing how to deal with the fact that Asia and Eurōpē join up on the north side of the Black Sea, where several Greek colonies had recently been established.

He chose a dividing line, one side was 'Europe' and the other side was 'Asia', and the names have stuck ever since. The main difference between then and now is the location of the dividing line. Some ancient geographers drew the dividing line at the Caucasus mountains, like the modern convention; others put the dividing line at the river Don, in the southwest corner of Russia.

Here's an answer I wrote back in May that goes into more detail about the origins of the name 'Asia'. The tl;dr is that it comes from the Bronze Age toponym Assuwa. As for 'Europe', we can theorise about its original location, but the etymology of the name is unknown. It's definitely not related to the mythical Lebanese princess Europa, who's the one that the moon of Jupiter is named after.

Like I said, this is the origin story, not an explanation of why the names stuck for so long in such a wide variety of historical contexts. That part of the story I'll have to leave to someone else.