This answer only considers the nuances of habere, not a comparison between it and the possessive dative.
The possessive genitive is different; it functions mostly like the English genitive and is used to express things like "my dog" rather than "I have a dog".
The example of the pope actually makes a good example for habere.
The canonical announcement upon ...
I think that the dative sibi can be crucially related to the presence of the preverb con-. A typical use of the dative case is found with some compound verbs and adjectives.
sī sibi ipse cōnsentit (Cic. Off. I.5) 'if he is in accord with himself'
So note that the the same dative you see with adjective conscius also appears with the related verb ...
Is this an example?
Cicero: sibi enim bene gestae, mihi conservatae rei publicae dat testimonium.
Perhaps it can be argued that sibi and mihi are datives of reference, but "agent" seems most natural to me. "He testifies that he performed good deeds, but that I preserved the republic." Unless a dative of reference is usual with testimonium, which I don't ...