I think a lot of languages use a possessive adjective rather than the genitive pronoun...
French certainly does ("mon chapeau" "ma maison"). Russian does. German does (come to think of it, I'm not sure there is an independent genitive form for the personal pronouns in German -- at least, I can't think of a situation where I would use one.)
This development is actually fairly logical, when you think about it. Unlike the other cases, the genitive almost always modifies another noun. A word that modifies another noun performs much the same function as an adjective. Since the pronouns are so common, why not just make an adjective out of them rather than saying "of me" "of him" (and so forth) all the time?
(When Latin does use a pronoun to indicate possession/ownership, it often prefers the dative rather than the genitive: "nomen mihi Marcus est" "est liber tibi")
IPHIGENIE: Kann uns zum Vaterland die Fremde werden?
ARKAS: Und dir ist fremd das Vaterland geworden.
IPHIGENIE: Das ist's, warum mein blutend Herz nicht heilt.
(Goethe, Iphigenie auf Tauris)