Are non-japanese people living in Japan well accepted for Japanese people?

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It depends mainly on the foreign person's mindset and what s/he thinks "being accepted" means.
In mono-racial countries like Japan, foreign-looking people are considered as foreigners however long they live there. Plus, western people's values and mindsets are drastically different from Japanese. As long as they think their way is the best and the only right, they wouldn't be able to fit in forever.