It's 2015. That's when Back to the Future: Part II takes place. And while we aren't driving flying cars and wearing double neckties, the movie did totally nail one point about Japan and, well, fax machines.
While faxes are a relic of the past for many, in 2015, people in Japan continue to use them -- something that both The Washington Post and The New York Times have reported in 2012 and 2013, respectively. I have a fax machine. Pretty much every Japanese person I know does as well. Heck, today, a company even asked me to send them a fax -- which I did.
Landlines are typically sold with fax machines, so even if people don't use them on a daily basis, they typically have them. With a fax infrastructure like that, you inevitably end up using them one way or another.
There are numerous reasons why the Japanese have stuck with the fax machine, with the biggest being perhaps that for older people they are easier to use than computers. Just ask Marty's boss, Fujitsu-san.