Idiomatic for me would be this simple, short, clear thing:

   def greeter(name):
     def greet():
       print(f"Hello {name}")
     return greet

   greet1 = greeter('Joe')
   greet2 = greeter('Sue')

   greet1()
   greet2()

Or else a class. I don't quite understand why you'd want a global in a module not mean the canonical 'this singletone thingie configures this module' ?