Phil Goddard has it right: this is a court order empowering someone to become the legal guardian of an incapacitated adult to whom they are related.
That said, calling it a "family guardianship" could make readers think the whole family is empowered to act as guardian. I would suggest "family-member guardianship," because it's normally one relative, sometimes (as needed) more than one, but not the entire family:
Art. 494-1: "le juge des tutelles peut habiliter une ou plusieurs personnes choisies parmi ses proches au sens du 2° du I de l'article 1er de la loi n° 2015-177 du 16 février 2015 à la représenter ou à passer un ou des actes en son nom..."
The "proches" under that law are: les "ascendants, descendants, frères et sœurs, partenaire d'un pacte civil de solidarité ou concubin..." (
https://www.legifrance.gouv.fr/affichTexteArticle.do?cidText... 
approbation des pairs (net) : -1
