Where exactly are you looking? On my install on my laptop, I see first the choice of either ‘Native’ or ‘SQL’ next to the Storage Type. When I click on ‘SQL’ I then get additional options appear for database driver, hostname, port, username & password. The driver list should show whatever drivers you have installed on the Portfolio Server. So for MySQL you would need to install MyODBC on your Portfolio Server (I’m running MyODBC 5.1), and then configure a DSN using the ODBC Administrator Tool. This is no longer included as part of OSX but is still available as a separate download. See this Apple KB for that. Then any DSNs you configure should appear when you fill in the appropriate fields and click to ‘Show Databases’, where you can then select the appropriate database and ‘Create’ the catalog.
Personally I’ve only ever done it with MySQL on the same machine, but given that just means I use localhost entries in my DSN, there’s no reason it shouldn’t work with remote MySQL servers, so long as it’s configured correctly, and they’re of a supported version. Do you know what version of MySQL is running on the QNAP/Synology server?