With the last change of forum system some years ago, the C++ plug-in developer forum was merged to here as "SDK" topic that this question also correctly applies. Just not seeing too much activity any more except for newbie shotgun posts.
That style dialog is a selectable dialog, similar to the prefs dialog.
The service ID kTextStyleParaSelectableService is missing in the SDK - I can only guess the original motivation for that decision decades ago, there is a whole family of related dialogs (e.g. find-change, control strip) and it is a bit complicated to support the text target server protocol used across all of them. You'd also implement your underlying feature as new attributes which has additional requirements. Doable with serious effort, but I'm not aware of an SDK example that would cover all of this out of the box.