If you have shared extension in different DBs, you need to edit the MySQL and change all the links to reflect the new DB names.
If you had re-use exactly the same DB name, this is not required.
The procedure that is given in the FAQ backup/restore suppose that keep the same DB name in the new server. Otherwise, you have to fix all the MySQL manually and replace all the old name by new name.
If you want that we help you doing such move, this is possible with our billable support that you can order at
www.jms2win.com/en/joomla-multisite?page...35&category_id=1
There is no magic solution. Mainly when the sharing is cross DBs.