This is a PhD in convention. Different institutions (currently Sapienza, Tor Vergata, INAF, ASI) provide the resources for fellowships, using both institutional and research funds. Thesis assignment rules are in place to ensure a distribution of PhD students supervised in the different institutions compatible with the provided resources.
In practice, once the students are enrolled, they are requested to provide a prioritized list of PhD thesis titles (selected from the full list of available theses for that cycle).
Once all the prioritized lists are received, the theses are assigned, starting from the first ranked in the admission procedure, and according to the expressed priority. The first choice is assigned if still available (not already selected by someone else) both in terms of thesis title and availability of fellowship in the institution. Otherwise, the second choice is assigned, and so on. For fellowships which are not supported by specific research funds, it is also checked if the supervisor already has another student to supervise in the same cycle.
Usually most of the PhD students obtain their first choice, but this is not granted, especially for the low-end of the ranking.