Not universally. Notice the widgets that have text selection seem like HTML. That's because they are. FX supports embedding HTML. But then it scrolls oddly.
It's configurable in JPro.
In JPro you can say, that the text-elements should support the standard browser text selection. This is the case for https://www.jpro.one/ . The demos usually have this feature disabled - most of the demos are more like "rich desktop applications".