The essential case where it makes sense for text to be non-selectable is on objects that can be dragged around. You definitely don't want to get the text selected when the user wanted to move its container.

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

That makes sense. I would say that at least draggable elements shouldn't be selectable on the web neither.

But should non-draggable elements in native apps be selectable?

> 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.