I need to manually set up for each of my WordPress sites a database. Since database requires a user with granted privileges to it my question is this: can I create one user and give it privileges to all databases of all sites or should every database have a separate user?
You CAN use just the one user and connect to multiple databases, however I'd advise not doing this. If someone was to compromise that username & password, they'd then be able to access all other sites.
Also, if you're going to be setting up multiple wordpress installations, you may want to check out the Wordpress MU (multi-user). This would mean you'd only need to keep one software up to date, but have multiple domains/wordpress configurations.
There's a product called "Wordpress MU". This is the same software that Wordpress developed for wordpress.org which allows multiple people to have their own individual blog - but there's only one core of files that need to be updated when there's new patches. Each of the users (sites) would use the core files for how the program works, but have their own individual templates for display etc. Databases are separate too.