I can understand your frustration with stereotypes and inaccurate labels, Samuel, but I think this topic is, by it's very nature, too impossibly broad to ever resolve satisfactorily.
You say you find the current labels of Europe and Asia inappropriate. What, then, would be appropriate? Discarding of continents altogether because of their lack of geographical seperation? Then what about the borders of individual countries? Why is the line between two countries in this particular place, as opposed to another? Is it fair to call one person Indian and another Pakistani if they live meters away from each other and share the same culture and language? It's just an arbitrary political line, isn't it? But then if you divide people based on culture, based on language and religion... very few places are isolated enough to categorise them effectively... there will always be blending and mixing and resulting in no clear divide between them. Now you can't group two different towns or villages together in one category because they consist of different people with different micro-cultures.
You can follow that train of thought down to individual people. Everyone is different. How does it make sense to categorise one person with another? They are individuals and quite seperate from each other in values and outlook.
How can you even begin to devise a label that would be appropriate? Any label is going to involve stereotyping, and stereotyping is going to involve misconceptions and assumptions.
People use the stereotypes of "Asian" and "European" and so on, because they are convenient ways to refer to a group of people. The speaker assumes the listener understands the group in the same way he does. Describing someone as "Asian" here (Australia) is shorthand for describing a person as having south-east Asian heritage (Malaysian, Singaporean, Indonesian, Filipino, etc). It makes no assumptions about religion, culture or even country. If necessary, the speaker can then go into greater detail about this person. Of course, the word 'Asian' has a different assumed meaning elsewhere (eg. the UK), but that is true of language in general.
Anyway, I think I'm just trying to say that I think you're going much too far in the point you're making - I agree that 'Asian' is used inaccurately, in the technical sense, but when you start arguing the validity of geopolitical borders, I think it's a different kettle of fish altogether.
The purpose of language is to provide accurate communication. If both parties understand the same thing, does it really matter that the specific definition of that word is somewhat different? (All you have to do is look at the roots of most words to see how the understood definitions have changed over time from the original 'official' meaning.)
Eg, when someone asks for a 'Kleenex', both parties understand that they mean a 'tissue', regardless of the brand. The fact that it was described as a Kleenex doesn't really matter.
If the point you are making is that the generalisation comes hand-in-hand with racism (or a subtle form of it), then that, too, is a different issue altogether.