Despite there being an option for Windows version of VLC, it seems to be absent in the OS X implementation of it. In order to run multiple instance of VLC on OS X you will have to open up Terminal and run the following command
This assumes that you installed VLC into the Applications folder. Otherwise you will have change it to the appropriate path where it is installed. It seems that running multiple instances of a program is discouraged by Apple’s usability guidelines (as mentioned by the forum poster) which would be ironic as I believe QuickTime files in Windows all open in a new window / instance as opposed to using an existing one.