In some countries there are not two American continents. In the US we use the 7 continent model, but not all countries do.
We do say the Asian continent. We don’t say The America, but we do say The Americas. We also say The Bronx, but not the Brooklyn. I have a feeling it just is what it is without a good explanation.
Although, maybe most of the English speaking countries use the 7 continent model? I don’t know.
Maybe since the US uses America, putting the The in front of The Americas, helps differentiate America from The Americas. Many years ago, and maybe still in parts of the world, people referred to America as the entire continent. My SIL’s Inlaws always called her The American (In Italian) and she was actually Mexican.
A song I have liked for a long time, Jose Luis Perales sings about “America” but if you listen to the words it seems he is talking most likely about Latin America.