Ok, so phpmyadmin is able to use the mysqli extension from php.
May you please try this:
1) Make a backup of the file /usr/local/ispconfig/interface/lib/config.inc.php
2) Change the mysql login details to user "root" and the mysql root password in that file and try if the ispconfig login works with that.
This way we can find out if its a user privilege problem of the mysql user.