I was thinking of writing a document about countries in Asia since most people seem to think only China & Japan are Asian, but I ran across some problems (Russia & Turkey). This made me become curious on how some countries are regarded as part of one continent but not another, and how some countries are even sometimes referred to as being in both continents (Eurasia) .
It isn't a geographical convention, obviously -- Eurasia is a single landmass -- but a story of how the names got attached to different parts of the one landmass.
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. Hecataeus, who was describing the entire world as known to him, was in the position of getting to choose how to deal with the fact that the two lands join up on the north side of the Black Sea.
As a result he chose a dividing line between 'Europe' and 'Asia', and the names have stuck ever since. The main difference between then and now is that we're not sure whether Hecataeus drew the dividing line at the Caucasus mountains or at the river Don. Other ancient geographers chose one or the other.
Since this is more about the names 'Europe' and 'Asia' than anything geographical, here's an answer I wrote back in May that talks about the origins of the name 'Asia'. The tl;dr is that it comes from the Bronze Age toponym Assuwa. The etymology of 'Europe' is unknown.