The reference manual you are quoting is not for python. I'm aware, that with gobject introspection, no python manual is available anymore (only a tutorial), but this manual does not document the python mapped functionality. A lot of functions from C have not been made accessible from other languages.
One way you might want to go, if you do not require Gio, is simply using python built-in functions?
concurrent.futures module does things in an async way (backported here).
With that you should be able to do something like this:
with concurrent.futures.ThreadPoolExecutor(max_workers=5) as executor:
executor.submit(shutil.copy, "test1.txt", "test1_new.txt").add_done_callback(
executor.submit(shutil.copy, "test2.txt", "test2_new.txt").add_done_callback(
More about this version here and here.