That's mostly a JS functionality from the looks of it. The little square icon should open a colour chooser which will apply the appropriate hex to the input fields. Otherwise you simply type it in.
The fields would be simply named something like fcolour and bcolour to see which is which and stored with the user in the table, and when you select you simply check if these fields exist and if they do you apply them to the name. The effect itself appears to be an old school glow filter that IE used to have, but I'd assume that in this day and age that a more cross platform compatible CSS could be used (but I haven't a clue you need to check with client developers about that).
Beyond that you need to do modifications to the forum itself to apply it during any display of username. How that's done completely depends on the system, it could be as easy as modifying a single function or object method during a fetch, to as complicated as modifying every script. For that you'd need to check the forums' forum for help.