The term distinction itself goes back to the Greeks, although it was never really rigorously defined. Asia essentially meant the place across the Aegean, which could refer to either Anatolia specifically or more broadly to everything we now call "Asia". Europe, on the other hand, basically meant the landmass Greece was on, which makes somewhat more sense if you understand that the early Geographers believed that the Black Sea directly connected to the Encircling Sea. With a more accurate geographic understanding the definitions are rather meaningless and incoherent, but they are ideologically useful, later morphing into a distinction between Catholic Europe and Orthodox and Muslim Asia, and then smoothly transitioning into the racialized understanding of today.

In my experience most scholars who do "deep history" like Ian Morris and Barry Cunliffe tend to call Europe a peninsula of the Eurasian continent.

EDIT: To add on to this, such confusion applied to many different geographical terms throughout antiquity. "Africa" for example sometimes meant essentially modern Tunisia, sometimes meant everything south of the Mediterranean. "Libya" usually meant everything between Egypt and "Africa" but often meant everything in Africa that wasn't Egypt. Same goes with Ethiopia. "India" is one of the stranger cases, originally referring to the Indus Valley, then the entire subcontinental peninsula, then everything beyond the Gulf of Aden.

Followup: Isn't Africa technically the same landmass (it's connected at the Sinai Peninsula, and very nearly connected at Gibraltar and Yemen)? Why has it been seen as distinct from Asia/Europe?