μου can be either a possessive determiner ("my") or a genitive pronoun ("to me" -- the old dative was replaced in the standard language by genitive in the personal pronouns).
So Ο νονός μου διαβάζει το μενού could be either Ο νονός μου - διαβάζει - το μενού "My godfather - reads - the menu" or Ο νονός - μου διαβάζει - το μενού "The godfather - reads (to) me - the menu".
If the second meaning is desired, then one spells the word μού as a signal that it is the personal pronoun and belongs with the following word, rather than the previous one.
It's mostly a spelling convention like telling η/ή, που/πού, πως/πώς etc. apart, though in speech the intonation and accentuation would also differ.
If you want to say "My godfather reads me the menu", you'd need μου twice: once for "my" and once for "me". And then I think you wouldn't write the second one μού since there can be no misunderstanding as to which role each of them serves.
That makes a lot of sense. The implied intonation from the accent seems to group the words together, and changes the flow of the pitch. Regarding using "..μου μου..", I guess that would be semantically redundant anyway, because if you are saying that a relative is reading the menu to you, specifying between "the godfather" and "my godfather" is unnecessary. If they were someone else's godfather, you would probably specify whose godfather they are, refer to them by name, or just as "he".
Ευχαριστώ πολύ mizinamo!
Always wondering, is ς a "s" sound or a "sh" sound?
Greek doesn't have a distinct "sh" sound, so Greeks don't have to be careful to keep their "s" and "sh" sounds separate -- as a result, their pronunciation of "s" can vary a bit more than it could in English. Sometimes, a Greek "s" will be somewhere in between a "typical" English "s" and a "sh", so it might sound more like a "sh" to you.
When you are speaking, I'd recommend using a "s" sound rather than the in-between sound you may hear from some Greeks.