Having a global len() function is very, very linked to polymorphism. You can call it on anything that has a __len__() function defined (i.e., any class that implements something like a "HasLength" interface in Java terms).
If you have __len__() implemented on the object, why not call it directly? If you are dealing with object X, that is of classes A,B,C or D (that all implement __len__), I don't see the fundamental difference between len(X) and X.__len__().
The "magic methods" wrapped in in double-underscores are special because other parts of the language's syntax rely on them always meaning the same thing. (Some room for programmer misbehavior here.)
Example: If __len__ is defined on an object x, and __nonzero__ (Py2) or __bool__ (Py3) is not defined, then "not x" will test if x.__len__() == 0.
Similarly, __contains__ interacts with "foo in x", and __eq__ and __hash__ are expected to play well together for hashable objects (e.g. unique members of a set).
The double underscores are a useful code smell that indicates you're changing how an object interacts with Python's syntax.
