Why do you make a difference between tabs in a native app and in a web app? The optimal UX should be the same.
Typically application tabs can be moved or recorded by dragging, and tabs in web pages can't; that would justify a different treatment. But it's because of the different behaviour of the tabs, not the different media
But should non-draggable elements in native apps be selectable?
Definitely yes. I hate it when I see an error message or a button label and I can't select the text to copy it for searching comments for it on the web.