I just recently decided to put the word "westerner" into wikipedia and it came up with the article "Western World". And it divided the use and meanings of Western into a few categories.

The historic West is originally the Mediterranean. Or in other words Greek and Roman culture. I guess Romans and Greeks were the first people to consider themselves Western. Which would make by this definition all people of Latin culture such as the French, Spanish, Mexicans, Brazilians, Portuguese and Italians more Western than Northern European Germanic people. Also this would make Turkey very Western because of being part of the Eastern Roman Empire.

Linguistically there's the Indo-European language family. And according to wikipedia, sometimes Hebrew as well. Because of the Abrahamic influence. Also there are Uralic languages spoken in Finland, Hungary and Estonia which are all considered part of the West.
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Japanese is a language isolate. Nobody knows for sure which language family Japanese belongs to. Though there are strong possibilities out there strongly supported by certain linguists.
* That Japanese is an Altaic language, related to the many North Asian languages. And can be extended even if further if the Ural-Altaic hypothesis is correct. That would make Japanese related to Finnish, Estonian and Hungarian. Maybe not Indo-European, but Western nonetheless. Heck, there's also the idea that Ural-Altaic languages are proto-Indo-European.
* That Japanese is part of the Eurasiatic language family, and distantly a part of the Indo-European family.
* If you consider Hebrew to be Western because of Christianity, many users have exerted similarities between Hebrew language and Japanese, claiming Japanese are part of the lost tribes of Israel. Not a well supported hypothesis among linguists, however.

Economically, being a first world country is considered Western. Japan is very high on the HDI, definitely a first world country. Heck, the Yen just surpassed the US dollar.

Politically, democracy is considered Western. Japan has become a democracy and is very anti-Communist generally. Shedding off the Fascism shared with Germany during WWII. Both countries evolving into the current political definition of Western equally.

Many consider Judeo-Christian religion to be Western. Coincidental or not, some users here have drawn similarities between Shinto and Abrahamic/Judaic religion. Again with that whole "lost tribes of Israel" thing.

In the end the East vs. West thing is a false dichotomy though. There is no such thing truly.