There’s no direct way to act on a selected “See Also” record but it’s possible to do the See Also comparison in AppleScript by using the compare command. You could then use the results to directly copy the name of e.g. the first result or show a list of all results.
This script shows a list and moves the selected record to the chosen “See Also” record’s parent group. You could use parts of it to do what you want. If you need help let me know.