I think it's because we ('the hoomans' ) like to compartmentalize stuff so it's easier to talk about them. Geographically speaking, there's a dozen or more continental plates (not accounting for the so called 'microplates'
which don't correspond to the land masses we've settled on.
North- and South America are also linked by a sliver of land and yet we tend to divide them into two separate continents. Which might not adhere to any straightforward definition (like 'a continent is a bunch of earth, elevated above sealevel, seperated from other continents by a bunch of water'
but it serves its linguistic purpose.
It's been a while since I was part of the education system (and oh boy it shows) but I do think that, linguistic definitions notwithstanding, kids still know that one can travel from Europe to Asia and back by land
.