Why was it impossible for Spain, or indeed any other European nation, to realize the extent of the territory Spain controlled after May 1493?

I think that "May 1493" refers to the papal bull "Inter caetera", which divided the world into two parts, giving the western part of the world (today almost all Americas except a small part of Brazil) to Spain.

At the time, the Americas were not yet mapped by the Europeans, so they did not know whether it was a small island or a huge continent - the correct answer is that the whole Americas have not yet been visited by the Europeans.