"The city of Rome" means not the city belonging to Rome, but the city that is Rome. Is there a term for this use of the possessive? Why do many languages use their possessive constructions for this non-possessive meaning? Is this usage restricted to specific families or geographical regions? Are there languages in which it's ungrammatical to do this? What are the restrictions on this construction (e.g. you obviously can't say "the president of Obama" to mean the president who is Obama)?
In Latin, this is called the genitivus explicativus, definitivus, appositivus or epexegeticus: the genitive defines or describes in other words what the head word is or consists in (not consists of). It exists in various Indo-European languages, possibly in all, including Latin, Greek, German, English, Dutch, French... In languages without cases, a preposition meaning "of" is normally used instead of / as the genitive. http://de.wikipedia.org/wiki/Genitiv
There is some connection with the genitivus partitivus, which describes what parts the head word consists of or includes: a box of matches, a cup of coffee, a regiment of archers.
Note that most cases and prepositions have widely different meanings. The preposition of in English has always been strongly associated with the older genitive case (which developed into 's). Similarly, many other languages also have a preposition they strongly associate with the genitive: Dutch van, German von, (late?) Latin de. This strong association, which may have been already present, was often strengthened as languages lost their cases. There is also cross-contamination between languages: the Latin use of the genitive and de may have influenced that of the Germanic languages in the Middle Ages. But some uses of the genitive were most probably already present in Proto-Indo-European, like possession. The same applies to other cases and prepositions, to varying degrees.
Definite descriptions such as "The City of Rome" are the subject of a 2013 paper by Michael Rieppel in Linguistics and Philosophy: 'The Double Life of the Mayor of Oakland'. Rieppel analyses descriptions conforming to the general schema: the + N + of + Proper Name, and draws a distinction between possessive descriptions on the one hand, as in (1), and identifying descriptions on the other, as in (2):
(1) The mayor of Oakland.
(2) The city of Oakland.
To answer your question then, one term used to pick out the class of definite descriptions you have identified is identifying definite descriptions. Rieppel describes the semantics of identifying descriptions as follows: the city of Oakland denotes the very same thing denoted by the proper name following of, and simply adds the information that it is a city. Rieppel goes on to provide several diagnostics showing that possessive descriptions can behave as semantic predicates, whereas identifying descriptions never do. Consider, for example, the fact that a possessive description can be coordinated with expressions that are uncontroversially predicative (3), whereas identifying descriptions cannot (4):
(3) She is ambitious, driven, and the mayor of Oakland.
(4) *It is lively, energetic, and the city of Oakland.
Rieppel points out that there is cross-linguistic variation in whether or not a prepositional linker such as of is present. In German, for example, identifying descriptions do not occur with a preposition:
(5) Der philosoph Frege The philosopher Frege
(6) Der Stadt Oakland The city Oakland
An interesting restriction on this construction is that the expression following of cannot just be any expression that denotes an individual, it has to be a proper name. Consider, for example, the contrast between (2) (repeated as (7)) and (8-10):
(7) The city of Oakland.
(8) *The city of that city.
(9) *The city of Agadir's sister city.
(10) *The city of the sister city of Agadir.
Even thought (8-10) all involve an expression that denotes an individual following of, they are nonetheless unacceptable. Rieppel develops an account of the semantics of identifying descriptions which arguably can derive these facts.