The root password of Ubuntu is never be asked when I use apt-get, which instead asking my user password. i checked my user and root password to the same when installation, if later i change user password for some security reason, would the root password change too?
Your user account and the root account are two different accounts. Changing the password for one, will not influence the other.
sudo tool you are likely using allows you to execute commands as root after entering your user password, at least it is usually set up that way for administrative users.
the root password is needed if you want to log in as root or start a terminal session as root via
In a GUI terminal session,
passwd will change your own password, while
sudo passwd or running
passwd in a root session can change root's password.