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I think we need to handle only two cases: short and fully qualified names. __qualname__ without __module__ doesn't make sense, and the value of tp_name depends on implementation details (is it Python class or builtin class, heap class or dynamic class?). Maybe use %t and %T?
But we may want to support formatting the name of the type itself and the name of the object's type. This give us 4 variants.
For old string formatting we can introduce new % codes (with possible modifiers). But in modern string formatting "T" can have meaning for some types (e.g. for datetime). We can implement __format__ for the type type itself (though it can cause confusion if cls.__format__() is different from cls.__format__(instance)), but for formatting the name of the object's type (as in your original proposition) we need to add a new conversion flag like "!r".