Does Kinoku require an account?
No. Kinoku has no sign-up or login, and no Kinoku server holds your training history. Android's own backup may still copy app data to your Google account.
No. You open Kinoku and start logging, with no sign-up or login. Kinoku does not run a cloud that keeps your training history, so your workouts, routines and stats live in the app on your phone.
Two things can involve your Google account, and neither one is a Kinoku account. Android system backup, when it is on, may copy eligible app data to your Google account. A Pro plan goes through Google Play, which handles the payment.
Without an account there is also no sync between devices. To move to a new phone, you can make a backup ZIP under Settings, then Import & Export, then Backup & Restore. An Android phone transfer that you start yourself carries your history across too.