The terms 'Europe', 'Asia', 'Libya' (later 'Africa') were used by the Greeks to denote regions of the world which were largely separated by water.

(Europe and Asia being separated by the Aegean Sea, the Straits and the Black Sea, Asia and Libya being separated by the Red Sea, Europe and Libya being separated by the Mediterranean).

These traditional names remained even when geography became better known, and it was apparent that Europe and Asia were more connected by land than separated by Seas.

Being connected by land does not really define what makes one continent rather than two (or we would have 'America', not 'South America' and 'North America') however, it is undoubtedly the impact of history and tradition which determines that we refer to Europe and Asia as separate continents.

There is no set official definition of 'continent'

If we're talking about purely geographical terms, technically Europe and Asia are just one continent - Eurasia (and technically Africa would be part of it too since it was connected by land continuously until the Suez Canal..)

Now if we're talking about cultural aspects, then Europe being it's own continent is more fair

However, as to why it's widely considered separate continents, it's mostly due to the Western Euro-centric view of the world which considers that west of the Ural mountains to be "European" and thus its own continent. Little known fact is that some European countries (mostly in Eastern Europe and Russia) and many parts of Asia consider Eurasia one continent when counting continents but will refer to Europe as its own separate cultural sphere