"Whose" is the only possessive relative pronoun in English. The antecedent of "whose" can be both people and things.
( - Purdue OWL)
"Whose" is not complete as a relative possessive pronoun. Its use favors people over abstract referents. "Democracy, whose core virtue is..." sounds artful and not standard.
Are there languages with a fully functional relative possessive pronoun? Is there a cross-linguistic reason why relative pronouns would be less likely to have a possessive form than other types of pronouns?
Possessive forms signify reference, as in "Japan's winters". A social concept of "ownership" is a subclass of that, as in "My car." It doesn't seem like the reason a relative possessive pronoun would favor entities with agency is because only agents are capable of consciously possessing something.
I believe the reason could be syntactic. It could have something to do with relative pronouns being able to take long ("unbounded") complements which constrains possessive pronouns to avoid... excessive syntactic content within their subtrees. Maybe it makes parsing sentences harder. (Or, it is a quirk of English where we only have relative pronouns which take a clause as complement, or possessive pronouns which take a noun phrase as complement, and there is some attribute of syntax that "blocks" having both.)