There is a significant difference between French and English which is causing some of the confusion here. In English, each personal pronoun has its own possessive form
I: my, you: your, he: his, she: her, it: its, they: their, one: one's etc.
French is different because all its third-person pronouns use the same possessive form
il, elle, on, ils, elles: son/sa/ses
as well as the obviously related same reflexive pronoun, se.
This means that the question in English is fundamentally different from the question in French.
From the etymological point of view, several languages have these s- pronouns, that are not well explained, popping up in different persons, including
German: sie (she, they, you)
Irish sinn (we) (as in Sinn Féin, We Ourselves) (Scots Gaelic similar)
Irish sé, sí, siad (he, she, they) (but not found in Scotland)
In French, there is a non-possessive s-pronoun: soi. This is the strong form of on and is apparently related to son/sa/ses (since soi/son/sa/ses corresponds to moi/mon/ma/mes). This means that son/sa/ses has a closer connection to a word for one than to to any other 3rd-person pronoun.