ModifySelf can modify what things?

In our RT Privileged users have rights to ModifySelf via System groups
permission for Privileged. When a user goes to Preferences -> About
me they can change everything on this page except for Organization,
which is blank. Can I set this globally and if not, how can I let
users edit this?

I’ve set
Set($Organization , “Organization Name”);
But that doesn’t populate the field. (Actually I have no idea what
this does but it sounded useful to set.)

Is this something only the Admin users can do? Since users can change
everything else about that page, it is confusing that this one field
is immutable.