Asia has very distinct countries and in some cases is even at war even if it's a cold one. Like India vs Pakistan, India vs China, North vs South Korea, China vs Taiwan. And customs, languages and (where applicable) religions are more radically different than within Europe too.
It makes less sense calling it "Asia" than it is calling Europe "Europe" :)
Not just that. If we get really pedantic, the EU is not only in Europe but includes territories in Africa (parts of Spain) and Asia (the entirety of Cyprus). And that's not even getting into the intercontinental shenanigans of France!