Right now posixpath returns the argument unchanged, ntpath performs a .replace(), and macpath a .lower(), so when non-string (or non-bytes) are passed to normcase the results are:
posixpath: arg returned as-is;
ntpath: AttributeError (object has no attribute 'replace');
macpath: AttributeError (object has no attribute 'lower');

In posixpath we could reject all the non-string (and non-bytes) args, raising a TypeError. For consistency, the other functions should raise a TypeError too, but I'm not sure it's worth changing it.

Attached a simple testcase that checks that normcase raises a TypeError for invalid values with all the three implementations.