I don't know if you went to a French-speaking school in Belgium, but I was told the same thing in a French school. The French "continent" certainly refers to a large mass of land and its surrounding islands [1]. Hence, the continent is Oceania rather than Australia.
That's how we learned it in middle school and beyond here in the US, but the Australia question boiled down to "yeah, technically it's an island, but are you really going to start your conversations about Australia with a bunch of quibbling around islands v. continents?".
40 years ago in Spain 5 continents only, no Antarctica, my guess is as no one was living there why worry about it. Don't know if it has changed, I'll ask my son.
North and South America are on different tectonic plates, while Europe and most of Asia are on a single plate. If anything, the geological definition of continents seems to support separate Americas and a unified Eurasia.
The best argument for separating the Americas is probably the existence of the Darien Gap. I’m not sure there is more inhospitable route in a populated area anywhere in the world.
The continents are defined geographically and culturally, not geologically.
https://en.wikipedia.org/wiki/Continent points out how they "are generally identified by convention rather than any strict criteria" and lists "several ways of distinguishing the continents", with the 7-continent model, two different 6-continent models. It also mentions the "four-continent model consisting of Afro-Eurasia, America, Antarctica, and Australia", as well as how there were only three discrete landmasses present during the Pleistocene ice ages, when the Bering Strait was instead land.
You can see the cultural influence in:
> In the English-speaking countries, geographers often use the term Oceania to denote a geographical region which includes most of the island countries and territories in the Pacific Ocean, as well as the continent of Australia.
> In some non-English-speaking countries, such as China, Poland, and Russia, Oceania is considered a proper continent because their equivalent word for "continent" has a rather different meaning which can be interpreted as "a major division of land including islands" (leaning towards a region) rather than "land associated with a large landmass" (leaning towards a landmass).
They are not defined by geology nor continental plates. For one, the word and current use is far older than our first glimmers of understanding plate tectonics. https://www.etymonline.com/word/continent says the meaning in the 1550s was "continuous tract of land" and by the 1610s became "one of the large land masses of the globe".
[1] https://www.larousse.fr/dictionnaires/francais/continent/186...