Yes, it is normal to log in each time to the Mi Flash Tool when you try to unlock your phone.
It is happening that because when you close the application the login session it is closed too. Also if you left the application running for a while your login session it is closed by the server, which is a security feature to ensure that a session it is authenticated again after a period of time.
I repeat it is abut only the Mi Flash Tool.
On the other hands, leave the current Mi Account on your phone, the message "do not add your account again in Miui again, otherwise you will wait from scratch" is referring only to leave on the phone your signed in Mi Account, do not remove it and add it again on the phone because this will reset the countdown period.