I think the OP is talking about different human identifications of what constitutes a certain continent, and not about the continents being physically different.
In any event, the Greeks called North Africa 'Libya' and the Romans called that region 'Africa', whereas sub-Saharan Africa was more commonly 'Aithiopia'/'Aethiopia'. They didn't always consider 'Asians' and 'Indians' to be one people. Their understanding of the further east was often hazy, but a definition of 'Asia' that stops at the Indus is absolutely plausible. 'India' could then be its own continent, and East Asia can very plausibly come to be considered a separate continent/geographic region. Meanwhile, the Greeks saw the far north of what we call Asia (the steppe, essentially) as distinct from Asia. Older sources (e.g. Homer) refer to the 'Cimmerian Steppe', but later on, 'Scythia' is of course also very possible. 'Europe', meanwhile, was north of the Med and west of Asia, but had no defined northern border (in paractice, the Greeks seemed to vaguely have the Alps in mind). This means that Northern Europe could conceivably get a name/identity of its own, distinct from 'Europe'. These definitions are completely cultural and have little to do with actualy continental plates, but could easily become deeply ingrained before plate tectonics are ever discovered, and thus remain the norm in cultural parlance.
The possibilities outside Eurasia are of course very broad, if we consider such an early divergence.