The Roster information presented in the selection box, and used for automatic selection of a Locomotive, is stored in a number of individual files. You can find their location by choosing "File Locations" from the main JMRI Help menu. Click the "Open Roster Location" button to open the directory in Finder/Explorer. More information is available on the "Configuration Files" Help page. You should only use the Roster menu to manipulate the Roster file, to ensure that it stays self-consistent.
To move the entire Roster to another computer, choose "Export Complete Roster" from the File menu, select a location to save the file, move that file to the other computer, then choose "Import Complete Roster" from the menu to import it there.
To do this manually at the file level, there will be a file called "roster.xml", and a directory called "roster". You need to move the "roster.xml" file, the "roster" directory, and all it's contents. Read more about the JMRI Configuration Files.