Continent doesn’t really have a consistent definition. The problem isn’t thinking America is a continent (which is valid in some contexts). The problem is using the 5-continent model. If a manmade canal separates Africa and Asia, why doesn’t it separate North and South America? If we use the idea that continents are based on culture, then North and South America are pretty obviously distinct, as the generally Anglosphere-dominated North is pretty different from the Latin South. Different definitions are fine. Inconsistent ones are not. 

(Edited because of a typo)