On Windows, you can press Ctrl+C to copy the contents of native message boxes. Admittedly not very discoverable. They should have added it to the System menu as well.
A general facility to select labels should use an additional modifier key, since otherwise unsophisticated users will tend to select labels by mistake and get confused. Labels are already actionable elements, in that clicking on them focuses the associated control, or even changes its state as in the case of check boxes and radio buttons. Another example are the labels of menu items and buttons. Adding selection when you move the mouse a bit while hitting a label is too finicky IMO. If the UI is also a touch interface, it would only get worse.