I'm not aware of any planned/possible way to do copy exactly as you would an HTML page with HTML tags, but there's a big project to make selection and copying much simpler and more powerful (design doc [1], in progress PR[2]).
With that PR it should be easy to just add one SelectionArea to the root of your app in order to make everything selectable, like it would be on a web page. You don't need SelectableText widgets. There's an example of this in the PR [3]. No built-in support for multimedia or rich text copying yet though.
Text input fields should already do copy/paste pretty much the same as native.