Why are most people in anime white or European-looking (instead of Japanese)? Anatomically speaking Europeans don't look like that either. But most of the people look more European than Japanese, at least it seems to me ... Maybe it's the round eyes? And the non-black hair in some cases?
Just like @senshin stated in his comment, for the Japanese people, their characters do look Asian.
There is a really nice article written about this on thesocietypages and as stated there:

Why do the Japanese draw themselves as white? You see that especially in manga and anime.
As it turns out, that is an American opinion, not a Japanese one. The Japanese see anime characters as being Japanese. It is Americans who think they are white. Why?  Because to them white is the Default Human Being.
[...]
Americans apply this thinking to Japanese drawings. But to the Japanese the Default Human Being is Japanese! So they feel no need to make their characters “look Asian”. They just have to make them look like people and everyone in Japan will assume they are Japanese – no matter how improbable their physical appearance.

In your question, you also said &quot;Maybe it's the round eyes? And the non-black hair in some cases?&quot;
The article above also pointed out something about that:

[...] the Default Human Being thing is so strong that lacking other clear, stereotyped signs of being either black or Asian she defaults to white.
When you think about it there is nothing particularly white about how anime characters look:

huge round eyes – no one looks like that, not even white people (even though that style of drawing eyes does go back to Betty Boop).
yellow hair – but they also have blue hair and green hair and all the rest. Therefore, hair colour is not about being true to life.
small noses – compared to the rest of the world whites have long noses that stick out.
white skin – but many Japanese have skin just as pale and white as most White Americans.


Of course, there are also exceptions. A nice example is indeed Dan Eagleman, as stated by @senshin. The moment they need to make somebody look non-Japanese, they will try to design the character more to the American/European stereotype, i.e.

Bigger noses
Blond hair/blue eyes/white
Speaks fluent English