Is there a word in the English language more ambiguous than "mine"?
"My house," can be a place that you own, a place that you rent, a place that you live, a place that you dream of owning.
"My girl," can be a daughter, an employee, a friend friend girlfriend, an intimate girlfriend, a wife, a dog, a boat, a horse you've bet on in the Kentucky Derby.
"My class," can be a group of students who take instruction from you, a group of students who take instruction at the same time in the same subject from the same teacher, people of the same socio-economic status of you, people who are on track to graduate at the same time as you, people who preform at the same level that you do.
Attach mine or yours to theirs to any word and it takes on a multitude of multiple meanings.
The grammatical possessive is more about the existence of a connection than the precise nature of that connection. The term "possessive" in grammar is somewhat deceptive. It seems to imply ownership, but it really only implies connection.
This ambiguity is multilingual. "Chez moi," in French, which literally means "my house" has multiple levels of meaning and the possessive in other languages can be every bit as ambiguous as it is in English.
Perhaps there is some language that has a less promiscuous possessive, where ownership and affiliation cannot be expressed with the same phrase, but I don't know of any. I suspect that it is possible to distinguish different kinds of possessives with more exacting language in almost any language where those distinctions exist, but I don't know that there are languages where there are not utterly mundane expressions that don't make those distinctions in the same sense that many Romance languages don't allow a noun to lack a grammatical gender (they lack a neuter human and/or neuter not human definite article, for example).
This is a bit surprising. The general tendency in language is for every conceivable possibility to be found in some language. Some languages are ergative, some are not, and some are split-ergative. Some languages have two grammatical genders, some more than two, some none. Some languages have SOV word order, some SVO, and so on. The languages noun classes, some do not. Some have subject verb agreement, others don't. Some languages are isolating, others inflected and others agglutinative or fusional. Some languages use base ten, other base twenty for their numbers. Some languages are tonal, others lack tone. But, I've never heard of a distinction between languages with narrow and broad possessives.