I am trying to build a site that very closely resembles another and so far so good with WordPress Multisite Networks. I am also using the plugin WP-FileBase - this plugin creates a database table per site (http://wordpress.org/support/topic/plugin-wp-filebase-download-manager-does-it-work-on-multisite) - is there any way for me to set it up so that the sites share the same database?
You are asking a question that is specific to a plugin and not WordPress. How the plugin functions depends entirely on the plugin author's design. Reading the article above, sure the plugin works in WP multsite, but as the author mentions extra works needs to be done to share that content among other mulisite sites. You should follow up on the plugin support forum for answers.