They still do that kind of thing in Japan. Indeed, the Japanese "era" is due to switch over when the reigning emperor's abdication becomes effective in May 2019.
It's different today though. While the historically famous Edo period lasted 265 years, today the periods just end when an emperors reign ends. They are also counting years (today, not sure if they did earlier). E.g. 2018 is 平成30年 (Heisei year 30).
You're mixing up eras and reigns. The "Edo period" refers to the years when the shogunate ruled from Edo (today's Tokyo), during which there were over a dozen (notionally) reigning emperors:
Thank you. Didn't know that :) I misinterpreted stuff there
It doesn't affect your daily life much any more though. Those eras used to change when super significant events happened but now they are simply tied to emperors. The same system still applies in the UK where laws are numbered by the year of the reign of the then-current monarch, but I doubt anyone outside the parliament and law profession pays any attention.